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Squaring fractions is one of the simplest calculations performed on fractions. Very similar to squaring integers, meaning you just multiply the numerator and denominator by itself. [1] X Research Source In some cases you have to reduce fractions before squaring to make the problem simpler. If you have not yet learned this skill, the following article will help you quickly understand the problem through an easy-to-understand overview.
Steps
Calculate the square of the fraction
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Understand how to square integers. When you see an exponent of two, you have to square the number. To square an integer, multiply the number by itself. [2] X Research Source Example:
- 5 2 = 5 × 5 = 25
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The method of squaring fractions is similar. To square a fraction, multiply the fraction by itself. In other words, multiply the numerator and denominator by itself. [3] X Research Source Example:
- ( 5 / 2 ) 2 = 5 / 2 × 5 / 2 or ( 5 2 / 2 2 ).
- Square each number will give the result ( 25 / 4 ).
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Multiply the numerator and denominator by itself. You can multiply the numerator or denominator first, the order doesn’t matter, as long as both numbers are squared. Simply put, start taking the multiplier numerator for itself, then denominating the multiplier for itself.
- The numerator is the term above the fraction, the denominator is the term below the fraction.
- For example: ( 5 / 2 ) 2 = ( 5 x 5 / 2 x 2 ) = ( 25 / 4 ).
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Simplify fractions to end the problem. When working with fractions, the final step is always to reduce the fraction, or convert the fraction to a mixed number. [4] X Research Source In the above example, 25/4 is a non-true fraction because the numerator is larger than the denominator.
- To convert to a mixed number, we divide 25 by 4 and get 6 times (6 x 4 = 24), odd 1. So the mixed number is 6 1 / 4 .
Calculating the square of a negative fraction
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Recognize the minus sign before the fraction. Negative fractions will have a minus sign in front. People often put parentheses around a negative fraction so you know the “–” sign is the sign of the term, not requiring you to do the subtraction. [5] X Research Sources
- Example: (– 2 / 4 )
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Multiply the fraction by itself. Square the fraction as usual by multiplying the numerator by itself, and then the denominator by itself. It’s simpler that you multiply that fraction by itself.
- For example: (– 2 / 4 ) 2 = (– 2 / 4 ) x (– 2 / 4 )
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When two negative numbers are multiplied together, the result is a positive number. When there is a minus sign, the whole fraction is negative. When you square a fraction, you multiply two negative numbers. Whenever two negative numbers are multiplied, the result is positive. [6] X Research Source
- Example: (-2) x (-8) = (+16)
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Remove the negative sign after squaring. After squaring the fraction, you have multiplied two negative numbers. You will get a positive value from this calculation. Remember to write the final result without the minus sign. [7] X Research Sources
- Continuing with the above example, the fraction received will be a positive number.
- (– 2 / 4 ) x (– 2 / 4 ) = (+ 4 / 16 )
- By convention, the “+” before a positive number is omitted. [8] X Research Sources
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Reduce fractions to their simplest form. The final step when performing any calculation on a fraction is to reduce it. Fractions don’t really have to be simplified to mixed numbers and then reduced.
- For example, ( 4 / 16 ) has a common factor of four.
- Divide numerator and denominator by 4: 4/4 = 1, 16/4= 4
- The fraction has been reduced to: ( 1 / 4 )
Use short and quick calculation
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Check if you can reduce the fraction before squaring it. Simplifying the fraction first will make it easier to square. Simplifying fractions means that you have to divide the numerator and denominator by the common factor until the number one is the only number that the numerator and denominator can be divisible by at the same time. [9] X Research Source Simplifying fractions first means you won’t have to reduce larger numbers after squaring.
- Example: ( 12 / 16 ) 2
- 12 and 16 are both divisible by 4. 12/4 = 3 and 16/4 = 4; so 12/16 can be reduced to3/4.
- Now you will square the fraction 3 / 4 .
- ( 3 / 4 ) 2 = 9 / 16 , which can no longer be reduced.
- To prove this, square the original fraction without reduction:
- ( 12 / 16 ) 2 = ( 12 x 12 / 16 x 16 ) = ( 144 / 256 )
- ( 144 / 256 ) has a common factor of 16. Dividing both numerator and denominator by 16 reduces the fraction to ( 9 / 16 ), which is the same result we get when we reduce the original fraction first. .
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Know when to postpone fractional reduction. When solving more complex equations, you may end up eliminating some of the factors. In this case, you should postpone reducing the fraction to make the problem easier to solve. Adding a factor to the above example illustrates what has just been said.
- Example: 16 × ( 12 / 16 ) 2
- Solve the squared number and cancel the common factor 16:
16* 12 /16* 12 / 16- Since there is an integer 16 and two 16s in the denominator, you can cross out ONE of the two 16s in the denominator.
- Rewrite the reduced equation: 12 × 12 / 16
- Simplify 12/16 by dividing by4 : 3/4
- Multiply: 12 × 3 / 4 = 36/4
- Divide: 36/4 = 9
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Understand how to calculate exponents quickly. Another way to solve the above example is to simplify the exponent first. The end result is the same, only the solution is different.
- Example: 16 * ( 12 / 16 ) 2
- Rewrite the expression as the square of the numerator and denominator: 16 * ( 12 2 / 16 2 )
- Suppress the exponent in the denominator:
16* 12 2 / 162- Imagine the first 16 has an exponent of 1:16 1 . Using the exponent rule in division, you subtract two exponents from each other. 16 1 /16 2 , resulting in 16 1-2 = 16 -1 or 1/16.
- You now have: 122/16
- Rewrite and reduce the fraction: 12*12 / 16 = 12 * 3 / 4 .
- Multiply: 12 × 3 / 4 = 36/4
- Divide: 36/4 = 9
Things you need
- Paper or writing board
- Pencil/ballpoint pen (for writing on paper)
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This article is co-authored by a team of editors and trained researchers who confirm the accuracy and completeness of the article.
The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.
This article has been viewed 38,489 times.
Squaring fractions is one of the simplest calculations performed on fractions. Very similar to squaring integers, meaning you just multiply the numerator and denominator by itself. [1] X Research Source In some cases you have to reduce fractions before squaring to make the problem simpler. If you have not yet learned this skill, the following article will help you quickly understand the problem through an easy-to-understand overview.
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