Our square root calculator estimates the square root of any positive number friend want. Just enter the chosen number and read the results. Every little thing is calculated quickly and automatically! with this tool, friend can also estimate the square the the desired number (just enter the value right into the second field) which might be a an excellent help in recognize perfect squares from the square root formula.Are girlfriend struggling through the simple arithmetic operations: including square roots, subtracting square roots, multiply square roots or separating square roots? Not any more! In the following text, you will discover a detailed explanation around different square root properties, e.g., just how to leveling square roots, with countless various examples given. V this article, friend will discover once and for all exactly how to find square roots!
Have you ever before wondered what is the beginning of the square root symbol √? We have the right to assure you the this background is not as basic as you might think in ~ first. The beginning of the root symbol goes earlier to ancient times, together the origin of the percent sign.
You are watching: Square root of 3 divided by 2
If you're looking for the square source graph or square root function properties, head directly to the proper section (just click the links above!). There, we describe what is the derivative that a square root utilizing a an essential square root definition; we likewise elaborate on how to calculation square root of index number or square root of fractions. Finally, if you are persistent enough, girlfriend will discover out that square source of a an adverse number is, in fact, possible. In the way, we introduce complex numbers i m sorry find large applications in physics and mathematics.
Square source symbol √
The procedure of the square root of a number was currently known in antiquity. The more quickly clay tablet computer with the correct worth of up to 5 decimal areas of √2 = 1.41421 comes from Babylonia (1800 BC - 1600 BC). Plenty of other documents present that square root were additionally used by the ancient Egyptians, Indians, Greeks, and Chinese. However, the beginning of the source symbol √ is still mainly speculative.many scholars think that square roots originate indigenous the letter "r" - the first letter the the Latin indigenous radix meaning root,another theory states that square root symbol was taken native the Arabic letter ج the was put in its original form of ﺟ in words جذر - source (the Arabic language is written from ideal to left).
The an initial use of the square source symbol √ didn't include the horizontal "bar" over the numbers within the square root (or radical) symbol, √‾. The "bar" is recognized as a vinculum in Latin, definition bond. Back the radical symbol through vinculum is currently in day-to-day use, we generally omit this overline in the numerous texts, favor in write-ups on the internet. The notation the the higher degrees that a root has been suggested by Albert Girard who placed the degree index within the opened of the radical sign, e.g., ³√ or ⁴√.
The last question is why is the square root operation called root regardless of that true origin? The explanation must become more evident if we compose the equation x = ⁿ√a in a different form: xⁿ = a. X is called a source or radical since it is the covert base the a. Thus, words radical doesn't mean far-reaching or extreme, yet instead foundational, reaching the root cause.
Square root definition
In mathematics, the classic operations on numbers room addition, subtraction, multiplication, and also division. Nonetheless, us sometimes add to this perform some much more advanced operations and manipulations: square roots, exponentiation, logarithmic functions and even trigonometric attributes (e.g., sine and cosine). In this article, we will emphasis on the square root an interpretation only.
The square root of a provided number x is every number y who square y² = y*y yields the original number x. Therefore, the square source formula deserve to be express as:
√x = y ⟺ x = y²,
where ⟺ is a mathematical prize that means if and only if. Each confident real number always has actually two square roots - the an initial is optimistic and second is negative. However, for countless practical purposes, we usually use the positive one. The only number that has one square source is zero. The is due to the fact that √0 = 0 and zero is neither optimistic nor negative.
There is also another usual notation of square roots that can be much more convenient in many complex calculations. This alternative square source formula says that the square source of a number is a number increased to the exponent the the portion one half:
√x = x^(1/2) = x^(0.5)
In geometric interpretation, the square source of a offered area that a square provides the size of that is side. That's why √ has actually word square in that name. A similar situation is v the cube root ∛. If you take the cube root of the volume that a cube, you get the length of that edges. When square roots are offered when considering surface ar areas, cube roots are helpful to determine quantities the relate come the volume, e.g., density.
How to discover the square root?
Maybe us aren't being really modest, but we think the the best answer to the question just how to find the square source is straightforward: use the square root calculator! You have the right to use it both on her computer and your smartphone to easily estimate the square root of a given number. Unfortunately, there are sometimes instances when you have the right to rely just on yourself, what then? come prepare for this, you must remember several basic perfect square roots:square source of 1: √1 = 1, due to the fact that 1 * 1 = 1;square root of 4: √4 = 2, since 2 * 2 = 4;square source of 9: √9 = 3, since 3 * 3 = 9;square source of 16: √16 = 4, since 4 * 4 = 16;square root of 25: √25 = 5, because 5 * 5 = 25;square root of 36: √36 = 6, since 6 * 6 = 36;square source of 49: √49 = 7, since 7 * 7 = 49;square root of 64: √64 = 8, since 8 * 8 = 64;square source of 81: √81 = 9, due to the fact that 9 * 9 = 81;square source of 100: √100 = 10, due to the fact that 10 * 10 = 100;square root of 121: √121 = 11, because 11 * 11 = 121;square source of 144: √144 = 12, since 12 * 12 = 144;
The above numbers space the simplest square roots since every time you obtain an integer. Shot to mental them! yet what have the right to you do once there is a number that doesn't have actually such a pretty square root? There are multiple solutions. First of all, girlfriend can try to predict the result by trial and also error. Let's say the you desire to estimate the square source of 52:You know that √49 = 7 and also √64 = 8 therefore √52 should be between 7 and also 8.Number 52 is closer come the 49 (effectively closer come the 7) so friend can shot guessing that √52 is 7.3.Then, friend square 7.3 obtaining 7.3² = 53.29 (as the square root formula says) which is higher than 52. You have actually to try with a smaller sized number, let's to speak 7.2.The square the 7.2 is 51.84. Currently you have actually a smaller number, yet much closer come the 52. If the accuracy satisfies you, friend can end estimations here. Otherwise, you can repeat the procedure through a number chosen between 7.2 and 7.3,e.g., 7.22 and also so on and also so forth.
Another method is come simplify the square root an initial and then usage the approximations of the element numbers square roots (typically rounded to 2 decimal places):square source of 2: √2 ≈ 1.41,square root of 3: √3 ≈ 1.73,square source of 5: √5 ≈ 2.24,square source of 7: √7 ≈ 2.65,square root of 11: √11 ≈ 3.32,square root of 13: √13 ≈ 3.61,square root of 17: √17 ≈ 4.12,square source of 19: √19 ≈ 4.34, etc.
Let's shot and discover the square root of 52 again. You have the right to simplify it come √52 = 2√13 (you will learn exactly how to simplify square root in the following section) and also then substitute √13 ≈ 3.61. Finally, make a multiplication √52 ≈ 2 * 3.61 = 7.22. The result is the very same as before!
You can inspect whether a number is element or not with our element number calculator. A element number is a organic number (greater than one) the can't be derived as a product of 2 smaller natural numbers. For example, 7 is a element number because you can gain it only by multiply 1 * 7 or 7 * 1. On the various other hand, number 8 is not prime, because you can form it by multiply 2 * 4 or 4 * 2 (besides product the 1 and 8 itself).
Square source calculator
In some situations, friend don't need to recognize the exact result of the square root. If this is the case, our square root calculator is the finest option to estimate the value of every square root you desired. Because that example, let's speak you want to recognize whether 4√5 is higher than 9. From the calculator, you understand that √5 ≈ 2.23607, so 4√5 ≈ 4 * 2.23607 = 8.94428. It is an extremely close to the 9, yet it isn't greater than it! The square source calculator gives the last value with relatively high accuracy (to 5 digits in above example). Through the far-ranging figure calculator, you deserve to calculate this an outcome to together many far-reaching figures as you want.
Remember that our calculator instantly recalculates numbers gone into into either of the fields. You can find what is the square root of a details number by pour it until it is full the first window or obtain the square of a number the you gone into in the 2nd window. The second option is handy in finding perfect squares the are crucial in many aspects of math and science. Because that example, if you enter 17 in the second field, girlfriend will find out the 289 is a perfect square.
In some applications the the square root, particularly those related to to scientific researches such as chemistry and also physics, the results are desired in clinical notation. In brief, solution in clinical notation must have actually a decimal point between the very first two non-zero numbers and also will be stood for as the decimal multiply by 10 raised to an exponent. For example, the number 0.00345 is composed as 3.45 * 10⁻³ in clinical notation, vice versa, 145.67 is composed as 1.4567 * 10² in scientific notation. The results obtained using the square source calculator have the right to be converted to clinical notation v the clinical notation calculator.
How to leveling square roots?
First, let's questioning ourselves which square roots deserve to be simplified. Come answer it, you must take the number i m sorry is after ~ the square root symbol and find its factors. If any type of of its components are square number (4, 9, 16, 25, 36, 49, 64 and also so on), climate you have the right to simplify the square root. Why space these numbers square? They can be respectively expressed as 2², 3², 4², 5², 6², 7² and so on. Follow to the square source definition, you can contact them perfect squares. We've gained a one-of-a-kind tool dubbed the aspect calculator which might be an extremely handy here. Let's take it a look at some examples:can you simplify √27? v the calculator stated above, you attain factors that 27: 1, 3, 9, 27. There is 9 here! This method you deserve to simplify √27.can you leveling √15? factors of 15 space 1, 3, 5, 15. There are no perfect squares in those numbers, for this reason this square root can't it is in simplified.
So, how to leveling square roots? To describe that, we will usage a handy square root property we have talked about earlier, namely, the alternate square root formula:
√x = x^(1/2)
We can use those two forms of square roots and switch in between them whenever we want. Particularly, we remember that strength of multiplication that two details numbers is tantamount to the multiplication that those certain numbers increased to the exact same powers. Therefore, we have the right to write:
(x * y)^(1/2) = x^(1/2) * y^(1/2) ⟺ √(x * y) = √x * √y,
How have the right to you use this knowledge? The argument of a square root is typically not a perfect square friend can easily calculate, yet it may contain a perfect square amongst its factors. In various other words, you can write it as a multiplication of 2 numbers, where among the numbers is the perfect square, e.g., 45 = 9 * 5 (9 is a perfect square). The need of having at least one factor the is a perfect square is important to simplify the square root. In ~ this point, you should probably recognize what the next step will be. You must put this multiplication under the square root. In our example:
√45 = 45^(1/2) = (9 * 5)^(1/2) = 9^(1/2) * 5^(1/2) = √9 * √5 = 3√5.
You have efficiently simplified your first square root! that course, friend don't need to write under all these calculations. As long as friend remember the square root is identical to the power of one half, you deserve to shorten them. Let's exercise simplifying square roots through some other examples:How to simplify square root of 27? √27 = √(9 * 3) = √9 * √3 = 3√3;How to leveling square root of 8? √8 = √(4 * 2) = √4 * √2 = 2√2;How to leveling square source of 144? √144 = √(4 * 36) = √4 * √36 = 2 * 6 = 12.
In the last example, you didn't have to simplify the square root at all, due to the fact that 144 is a perfect square. You could just remember that 12 * 12 = 144. However, we want to present you that with the process of simplification, girlfriend can quickly calculate square root of perfect squares too. It is valuable when dealing with huge numbers.
Finally, you might ask exactly how to simplify roots of greater orders, e.g., cube roots. In fact, the process is very analogical to the square roots, however in the instance of cube roots, you have to discover at least one factor that is a perfect cube, no a perfect square, i.e., 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³ and also so on. Then you divide your number into two parts and also put under the cube root. Let's take the following example of simplifying ³√192:
∛192 = ∛(64 * 3) = ∛64 * ∛3 = 4∛3
It may seem a little bit facility at an initial glance, however after some practice, you will be able to simplify roots in your head. Trust us!
Adding, subtracting, multiplying and dividing square roots
Adding square roots and subtracting square roots
Unfortunately, including or subtracting square roots are not as simple as adding/subtracting continual numbers. For example, if 2 + 3 = 5, it doesn't median that √2 + √3 equals √5. That's wrong! To know why is that, imagine that you have actually two different types of shapes: triangles 🔺 and circles 🔵. What happens when you add one triangle come one one 🔺 + 🔵? Nothing! you still have one triangle and one circle 🔺 + 🔵. ~ above the various other hand, what happens once you shot to include three triangle to 5 triangles: 3🔺 + 5🔺? You'll we gain eight triangles 8🔺.
Adding square roots is very comparable to this. The result of adding √2 + √3 is quiet √2 + √3. Girlfriend can't leveling it further. That is a various situation yet when both square roots have actually the very same number under the root symbol. Then we can include them just as continual numbers (or triangles). For instance 3√2 + 5√2 equals 8√2. The same thing is true subtracting square roots. Let's take a watch at more examples illustrating this square source property:What is 6√17 + 5√17? Answer: 6√17 + 5√17 = 11√17;What is 4√7 - 7√7? Answer: 4√7 - 7√7 = -3√7;What is 2√2 + 3√8? Answer: 2√2 + 3√8 = 2√2 + 6√2 = 8√2, due to the fact that we streamlined √8 = √(4 * 2) = √4 * √2 = 2√2;What is √45 - √20? Answer: √45 - √20 = 3√5 - 2√5 = √5, because we streamlined √45 = √(9 * 5) = √9 * √5 = 3√5 and √20 = √(4 * 5) = √4 * √5 = 2√5;What is 7√13 + 2√22? Answer: 7√13 + 2√22, we can't leveling this further;What is √3 - √18? Answer: √3 - √18 = √3 - 3√2, we can't simplify this further than this, but we at least simplified √18 = √(9 * 2) = √9 * √2 = 3√2.
Multiplying square roots and dividing square roots
Now, when adding square roots is a piece of cake because that you, let's go one step further. What around multiplying square roots and also dividing square roots? Don't it is in scared! In fact, you already did it during the great of simple square roots. Multiply square root is based on the square root residential or commercial property that we have actually used before a few times, the is:
√x = x^(1/2)
Do you remember exactly how to multiply numbers that are increased to the same power? as a reminder:
xⁿ * yⁿ = (x * y)ⁿ,
x^(1/2) * y^(1/2) = (x * y)^(1/2) ⟺ √x * √y = √(x * y).
As protest to addition, you deserve to multiply every 2 square roots. Remember that multiplication has commutative properties, that way that the order to which two numbers are multiplied does not matter. Few examples need to clarify this issue:What is √3 * √2? Answer: √3 * √2 = √6;What is 2√5 * 5√3? Answer: 2√5 * 5√3 = 2 * 5 * √5 * √3 = 10√15, due to the fact that multiplication is commutative;What is 2√6 * 3√3? Answer: 2√6 * 3√3 = 2 * 3 * √6 * √3 = 6√18 = 18√3, we streamlined √18 = √(9 * 2) = √9 * √2 = 3√2.
Dividing square source is practically the very same since:
x^(1/2) / y^(1/2) = (x / y)^(1/2) ⟺ √x / √y = √(x / y).
All you should do is to replace multiplication sign with a division. However, the division is no a commutative operator! You have to calculate the numbers the stand before the square roots and numbers under the square roots separately. Together always, some valuable examples:What is √15 / √3? Answer: √15 / √3 = √5;What is 10√6 / 5√2? Answer: 10√6 / 5√2 = (10 / 5) * (√6 / √2) = 2√3;What is 6√2 / 3√5? Answer: 6√2 / 3√5 = (6 / 3) * (√2 / √5) = 2√(2/5) = 2√(0.4), us switched there from a simple portion 2/5 to the decimal fraction 2/5 = 4/10 = 0.4.
Square root of exponents and fractions
Calculating the square source of the exponent or square source of the fraction might no be clear for you. However with the knowledge you acquired in the previous section, friend should uncover it simpler than girlfriend expected! Let's start with the square roots of exponents. In the case, it will certainly be simpler for friend to usage the alternative kind of square source √x = x^(1/2). Execute you psychic the power rule? If not, right here is a quick reminder:
(x^n)^m = x^(n*m),
where n and also m are any real numbers. Now, when you location 1/2 instead of m you'll obtain nothing else yet a square root:
√(x^n) = (x^n)^(1/2) = x^(n/2),
and that's exactly how you discover the square root of one exponent. Speak of exponents, above equation watch very comparable to the typical normal distribution density function, i beg your pardon is widely supplied in statistics.
If you're still not sure about taking square root of exponents, below are a couple of examples:square root of 2^4: √(2^4) = (2^4)^(1/2) = 2^(4/2) = 2^2 = 4,square root of 5^3: √(5^3) = (5^3)^(1/2) = 5^(3/2),square source of 4^5: √(4^5) = (4^5)^(1/2) = 4^(5/2) = (2^2)^(5/2) = 2^5 = 32.
As you deserve to see, periodically it is impossible to obtain a pretty result like the first example. However, in the 3rd example, we confirmed you a little trick through expressing 4 as 2^2. This strategy can frequently simplify more complicated equations.
What about square root of fractions? take a look in ~ the previous ar where us wrote around dividing square roots. Friend can uncover there the complying with relation that should explain everything:
(x / y)^(1/2) ⟺ √x / √y = √(x / y),
where x / y is a fraction. Below you can discover some examples of square root of a fraction:square root of 4/9: √(4/9) = √4 / √9 = 2/3,square source of 1/100: √(1/100) = √1 / √100 = 1/10,square root of 1/5: √(1/5) = √1 / √5 = 1/√5 = √5/5.
Leaving root in the denominator is no a very good habit. That's why we eliminated it in the last example. We simply multiplied both the numerator and denominator through the exact same number (we can constantly do that, together the number we multiply by equates to 1), in this situation by √5.
Square root role and graph
Functions pat a crucial role not only in mathematics but in many other locations like physics, statistics, or finance. Role f(x) is nothing an ext than a formula the says how the value of f(x) transforms with the dispute x. To watch some examples, check out our finance tools made by financial specialists, because that example, the compound attention calculator or future value calculator. Friend will uncover there some attributes that you can apply in actual life. They're a good help if you want to know how to calculation the compound interest or to estimate the future value of one annuity.
Below friend can find the square root graph, comprised of half of a parabola. Examine it and shot to validate, because that example, whether the square root function of x = 9 is 3 and also of x = 16 is 4 (as it should be).
You most likely have currently noticed the the square source of the area that a square provides its next length. This function is supplied in among our building and construction calculators - square footage calculator. If you setup to do any kind of renovation in the future, these tools could be a an excellent help. Don't forget to use them!
Derivative that the square root
A derivative the a role tells united state how fast this role changes v its argument. Among the simplest instances in physics is the position of an object and its velocity (the rate of change of position). Let's say the the role x(t) describes how the street of the moving automobile from a particular point transforms with time t. Perform you know what determines how quick the readjust is in your street traveled? The answer is the rate of the car! therefore the derivative the the position x(t) is velocity v(t)(velocity have the right to depend top top time too). To represent derivative, we normally use apostrophe v(t) = x'(t) or the derivative price v(t) = dx(t)/dt.
The derivative of the general role f(x) is not constantly easy to calculate. However, in part circumstances, if the function takes a specific form, we've obtained some formulas. Because that example, if
f(x) = x^n,
where n is any real number, the derivative is together follows:
f'(x) = n * x^(n-1).
It may not look at like, yet this answers the question what is the derivative that a square root. Carry out you mental the alternate (exponential) form of a square root? Let us remind you:
√x = x^(1/2).
You have the right to see the in this case n = 1/2, therefore the derivative the a square source is:
(√x)' = (x^(1/2))' = 1/2 * x^(-1/2) = 1/(2√x).
Since a number to a an unfavorable power is one over the number, the estimate of the derivation will certainly involve fractions. We've obtained a tool that can be essential when including or subtracting fractions with different denominators. That is called the LCM calculator, and it speak you exactly how to discover the Least typical Multiple.
The derivative the a square source is necessary to achieve the coefficients in the so-called Taylor expansion. Us don't desire to dive into details as well deeply, so, briefly, the Taylor collection allows you to approximate various functions v the polynomials that are much less complicated to calculate. Because that example, the Taylor growth of √(1 + x) about the allude x = 0 is offered by:
√(1 + x) = 1 + 1/2 * x - 1/8 * x² + 1/16 * x³ - 5/128 * x⁴ + ...,
which is valid for -1 ≤ x ≤ 1. Back the above expression has actually an infinite number of terms, to obtain the approximate worth you can use simply a few first terms. Let's try it! v x = 0.5 and very first five terms, you get:
√(1.5) = 1 + 1/2 * 0.5 - 1/8 * 0.25 + 1/16 * 0.125 - 5/128 * 0.0625,
√(1.5) ≈ 1.2241,
and the actual value, provided by our calculator, is √(1.5) ≈ 1.2247. Near enough!
That was a many maths and also equations so far. Because that those of friend who room persistent enough, we've all set the next section which describes how to calculate the square root of a an adverse number.
Square root of a an unfavorable number
At school, friend probably have been taught the square root of a an adverse number does no exist. This is true as soon as you think about only real numbers. A long time ago, come perform progressed calculations, mathematicians had actually to present a an ext general collection of numbers - the complex numbers. They deserve to be to express in the adhering to form:
x = a + b*i,
where x is the complex number through the real part a and imaginary component b. What differs between a facility number and a actual one is the imagine number i. Right here you have actually some examples of complicated numbers: 2 + 3i, 5i, 1.5 + 4i, 2. You might be surprised see 2 there which is a genuine number. Yes, the is, however it is also a complex number through b = 0. Complex numbers room a generalization that the genuine numbers.
So far imaginary number ns is most likely still a secret for you. What is it in ~ all? Well, back it might look weird, that is characterized by the following equation:
i = √(-1),
and that's all that you should calculate the square source of every number, whether it is confident or not. Let's see some examples:square source of -9: √(-9) = √(-1 * 9) = √(-1)√9 = 3i,square root of -13: √(-13) = √(-1 * 13) = √(-1)√13 = i√13,square source of -49: √(-49) = √(-1 * 49) = √(-1)√49 = 7i.
Isn't the simple? This trouble doesn't arise v the cube root since you can achieve the an unfavorable number by multiplying 3 of the identical an unfavorable numbers (which girlfriend can't do with two an unfavorable numbers). For example:
³√(-64) = ³√<(-4)*(-4)*(-4)> = -4.
That's probably whatever you should know around square roots. We appreciate that you remained with us until this point! together a reward you should bake other sweet for yourself :-) examine out our perfect pancake calculator to find out how to do the perfect pancake, yet you favor it. You may need ours grams to cups calculator to assist you with this. It works both ways, i.e., to transform grams to cups and convert cup to grams. And if you ask you yourself "How countless calories need to I eat a day?", visit our comfortable calorie calculator!
Can a number have more than one square root?
Yes, in fact all positive numbers have 2 square roots, one the is positive and also another the is same but negative to the first. This is because if girlfriend multiply two negatives together, the negative cancel and also the result is positive.
How carry out you discover the square source without a calculator?Make one estimate the the square root. The closest square number is agree if you’re at a loss.Divide the number you want to uncover the square source of through the estimate.Add the calculation to the result of action 2.Divide the result of step 3 through 2. This is your brand-new estimate.Repeat procedures 2-4 v you brand-new estimate. The an ext times this is repeated, the more accurate the result is.
How can I calculation square roots?Find the nearest square number above and below the number you are thinking of.The square root will certainly be in between the square root of these numbers.The closeness of the number come a square source indicates how close the source is. E.g., 26 is an extremely close to 25, therefore the root will certainly be really close to 5.Try a couple of times to obtain the hang of it.
Is the square source of 2 a rational number?
No, the square source of 2 is not rational. This is because when 2 is composed as a fraction, 2/1, it can never have actually only also exponents, and therefore a reasonable number cannot have actually been squared to produce it.
How can I get rid of a square root?
In algebra, squaring both sides of the equation will obtain rid of any type of square roots. The an outcome of this operation is that the square roots will certainly be changed with everything number they to be finding the square source of.
Are square root rational?
Some square roots room rational, whereas others room not. You have the right to work the end if a square source is reasonable or no by finding out if the number you room square rooting deserve to be expressed in terms of only even exponents (e.g. 4 = 22 / 1 2). If the can, its source is rational.
Is the square root of 5 a rational number?
The square source of 5 is not a rational number. This is since 5 can not be expressed as a portion where both the numerator and also denominator have even exponents. This method that a reasonable number cannot have been squared to get 5.
Is the square source of 7 a reasonable number?
The result of square rooting 7 is an irrational number. 7 can not be created as a portion with only even exponents, an interpretation that the number squared to reach 7 cannot be expressed together a fraction of integers, and also therefore is no rational.
What is the derivative of the square source of x?
The derivative that square source x is x-1/2/2, or 1/2SQRT(x). This is because the square root of x can be expressed as x1/2, from which differentiation occurs normally.
How carry out you uncover the square root of a decimal?Convert the decimal into a fraction.Find any kind of square roots of the fraction, or calculation it. Make the fraction equal come the square root you found squared.Cancel the square root and also the square leaving you v the fraction.
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Rewrite the portion as a decimal together your final answer.