Squaring fractions is one of the simplest things you can do. The procedure is very similar to the one used with whole numbers, because you just need to multiply both the numerator and the denominator by itself. There are cases in which it is better to simplify the fraction before raising it to a power, to make operations easier. If you haven't mastered this skill yet, this article will help you internalize it quickly.
Steps
Part 1 of 3: Squaring Fractions
Step 1. Learn how to raise integers to the second power
When you see an exponent of 2, you know you need to square the base. In case the base is an integer, just multiply it by itself. Eg:
52 = 5 × 5 = 25.
Step 2. Keep in mind that the procedure for squaring fractions follows the same criterion
In this case, just multiply the fraction by itself. Alternatively, you can multiply both the numerator and the denominator by themselves. Here is an example:
- (5/2)2 = 5/2 × 5/2 or (52/22);
- Squaring each number you get: (25/4).
Step 3. Multiply the numerator and denominator by themselves
The order in which you proceed is not important as long as you remember to multiply both numbers. To simplify the calculations, start with the numerator: multiply it by itself. Then repeat the process with the denominator.
- The numerator is the number that is above the fraction line, while the denominator is the one that is below.
- Eg: (5/2)2 = (5 x 5/2 x 2) = (25/4).
Step 4. Simplify the fraction to finish the operations
When working with fractions, the last step is to reduce the result to the simplest form or to turn an improper fraction into a mixed number. If you always consider the previous example, 25/4 it is actually an improper fraction, because the numerator is greater than the denominator.
To convert it to a mixed number, divide 25 by 4 and you get 6 with the remainder of 1 (6x4 = 24). The final mixed number is: 6 1/4.
Part 2 of 3: Square Fractions with Negative Numbers
Step 1. Recognize the negative sign in front of the fraction
When working with numbers below zero, you can see the minus sign ("-") in front of them. It is worth getting into the habit of putting the negative number in parentheses to remember that the "-" sign refers to the number itself and not to the subtraction operation.
Eg: (-2/4).
Step 2. Multiply the fraction by itself
Raise it to the second power, as you normally would, by multiplying the numerator and denominator by themselves. Alternatively, you can multiply the whole fraction by an identical one.
Here is the example: (-2/4)2 = (–2/4) x (-2/4).
Step 3. Remember that two negative factors generate a positive product
When the minus sign is present, the whole fraction is negative. When you square it, you are multiplying two negative numbers together which will result in a positive value.
For example: (-2) x (-8) = (+16)
Step 4. Remove the minus sign after squaring the fraction
When you do this, you are actually multiplying two negative numbers together. This means that the square of the fraction is a positive value. Remember to write the final result without the negative sign.
- Always considering the previous example, the final fraction will be positive:
- (–2/4) x (-2/4) = (+4/16);
- By convention, the "+" sign is omitted in front of numbers greater than zero.
Step 5. Reduce the fraction to its lowest terms
The last step you need to do in the calculations is to simplify the fraction. The improper ones must be transformed into mixed numbers and then simplified.
- Eg: (4/16) has the number 4 as a common factor;
- Divide the fraction by 4: 4/4 = 1, 16/4 = 4;
- Rewrite the fraction in simplified form: (1/4).
Part 3 of 3: Taking advantage of simplifications and shortcuts
Step 1. Check if you can simplify the fraction before squaring it
Generally, it is easier to reduce the fraction to its lowest terms before proceeding with elevation. Remember that simplifying a fraction means dividing the numerator and denominator by a common factor until they become prime to each other. If you do this first, it means you won't have to do it when the numbers are bigger.
- Eg: (12/16)2;
- 12 and 16 can both be divided by 4: 12/4 = 3 and 16/4 = 4; so 12/16 simplifies to 3/4;
- At this point, you can raise the fraction 3/4 squared;
- (3/4)2 = 9/16 which cannot be simplified any further.
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To verify these calculations, square the original fraction without reducing it to the lowest terms:
- (12/16)2 = (12 x 12/16 x 16) = (144/256);
- (144/256) has the number 16 as its common factor. Divide both the numerator and the denominator by 16 and you get (9/16), the same fraction that you calculated starting from the simplification.
Square Fractions Step 11 Step 2. Learn to recognize cases where it is best to wait before simplifying the fraction
When you have to work with more complex equations, you might just cancel one of the factors. In this case, it is easier to wait before reducing the fractions to a minimum. Adding one more factor to the previous example will clarify this concept.
- For example: 16 × (12/16)2;
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Expand the power and cancel the common factor 16: 16 * 12/16 * 12/16;
Since there is only one integer 16 and two 16s in the denominator, you can only delete one;
- Rewrite the simplified equation: 12 × 12/16;
- Simplify 12/16 dividing the numerator and denominator by 4: 3/4;
- Multiply: 12 × 3/4 = 36/4;
- Divide: 36/4 = 9.
Square Fractions Step 12 Step 3. Learn how to use the power shortcut
Another method of solving the same equation from the previous example is to simplify the power first. The final result does not change, because it is just a different calculation technique.
- For example: 16 * (12/16)2;
- Rewrite the equation with the power in the numerator and denominator: 16 * (122/162);
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Eliminate the exponent of the denominator: 16 * 122/162;
Imagine that the first 16 has exponent equal to 1: 161. Using the power division rule, you can subtract the exponents: 161/162 leads to 161-2 = 16-1 that is 1/16;
- You are now working with this equation: 122/16;
- Rewrite and reduce the fraction to the lowest terms: 12*12/16 = 12 * 3/4;
- Multiply: 12 × 3/4 = 36/4;
- Divide: 36/4 = 9.
