How to Square a Fraction With a Variable

By Kathryn White
Slope problems start with basic concepts and work toward more advanced skills.
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Squaring a number, or algebraic expression that contains a variable, means multiplying it by itself. Squaring numbers can be done in your head or on a calculator to get an actual answer, while squaring algebraic expressions is part of simplifying them. Squaring fractions with both numbers involves squaring the numerator and putting it in the numerator of the answer as well as squaring the denominator to put the result in the new denominator. Squaring fractions with variables in them works the same way, although there are certain expressions, such as binomials, that make the problems more difficult.

Method 1

Simplify the fraction by reducing the numbers and using the division exponent rule by subtracting the exponents for the variables that are like bases. For example, ((20x^6r^4)/(15x^2r^6))^2 would become ((4x^4)/(3r^2))^2.

Rewrite the problem as the fraction multiplied by itself. For example, you would rewrite (4x^4/3r^2)^2 as (4x^4/3r^2)(4x^4/3r^2).

Multiply the numbers in the two numerators together and the numbers in the two denominators together and apply the multiplication exponent rules to the variables by adding exponents of like bases. Here, you would end up with (16x^8)/(9r^4).

Method 2 - Applying the Square First

Simplify the number part of the fraction if possible. For example, you would change ((20x^6r^4)/(15x^2r^6))^2 to ((4x^6r^4)/(3x^2r^6))^2.

Multiply the exponent of 2 by each exponent inside the fraction and apply it to the numbers. ((4x^6r^4)/(3x^2r^6))^2 becomes (16x^12r^8)/(9x^4r^12).

Apply your division and multiplication exponent rules by subtracting or adding the exponents of like bases to simplify the fraction. For instance, (16x^12r^8)/(9x^4r^12) would end up as (16x^8)/(9r^4).

About the Author

Kathryn White has over 11 years of experience tutoring a range of subjects at the kindergarten through college level. Her writing reflects her instructional ability as well as her belief in making all concepts understandable and approachable. White earned a Bachelor of Arts in history from Illinois Wesleyan University.