9.5 Divide square roots

Divide square roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.

We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says

Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.

We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.

We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.

We will use the Quotient Property for Exponents, $\frac{a^m}{a^n}=a^{m-n}$ , when we have variables with exponents in the radicands.

Rationalize a one term denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. [link] shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.