9.5 Divide Square Roots

Square roots are essential in algebra, letting us solve equations and simplify expressions. Dividing square roots follows specific rules, like dividing radicands when indices match. We can also rationalize denominators to remove square roots from the bottom of fractions.

Mastering these techniques is crucial for working with more complex algebraic expressions. By understanding how to divide and simplify square roots, you'll be better equipped to tackle advanced math problems and real-world applications involving square roots.

Division and Simplification of Square Roots

Division with square roots

• Divide square roots with the same index by dividing the radicands using the formula $\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$ ($\sqrt{18} \div \sqrt{2} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$)
• Rewrite square roots with different indices to have a common index before dividing the radicands ($\sqrt[3]{8} \div \sqrt{4} = \sqrt[6]{8} \div \sqrt[6]{4} = \sqrt[6]{\frac{8}{4}} = \sqrt[6]{2}$)
• Dividing a square root by itself results in 1 using the property $\sqrt{a} \div \sqrt{a} = 1$
• Divide a number by a square root by writing the number as a square root and applying division rules ($4 \div \sqrt{2} = \sqrt{16} \div \sqrt{2} = \sqrt{\frac{16}{2}} = \sqrt{8} = 2\sqrt{2}$)
• The square root symbol (√) is used to represent the square root of a number

Rationalizing single-term denominators

• Rationalize a denominator containing a single square root term by multiplying numerator and denominator by the square root term using $\frac{a}{\sqrt{b}} = \frac{a \cdot \sqrt{b}}{\sqrt{b} \cdot \sqrt{b}} = \frac{a\sqrt{b}}{b}$ ($\frac{2}{\sqrt{3}} = \frac{2 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{2\sqrt{3}}{3}$)
• Combine like terms and reduce the resulting fraction if possible to simplify the expression ($\frac{3}{\sqrt{12}} = \frac{3 \cdot \sqrt{12}}{\sqrt{12} \cdot \sqrt{12}} = \frac{3\sqrt{12}}{12} = \frac{\sqrt{3}}{2}$)
• The denominator is the bottom part of a fraction, which is being rationalized in this process

Techniques for two-term denominators

• Rationalize a denominator with two terms by multiplying numerator and denominator by the conjugate of the denominator ($a + \sqrt{b}$ has conjugate $a - \sqrt{b}$) using $\frac{c}{a + \sqrt{b}} \cdot \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{c(a - \sqrt{b})}{a^2 - b}$ ($\frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$)
• Combine like terms and reduce the resulting fraction if possible to simplify the expression ($\frac{2}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} = \frac{2(1 - \sqrt{5})}{1 - 5} = \frac{2 - 2\sqrt{5}}{-4} = \frac{\sqrt{5}}{2} - \frac{1}{2}$)
• The numerator is the top part of a fraction, which is also affected during the rationalization process

Additional Concepts in Square Root Division

• Fractions are often involved in square root division, as they represent the division of two numbers or expressions
• Exponents play a role in simplifying square roots, as perfect square factors can be extracted (e.g., $\sqrt{16x^2} = 4x$ when $x$ is positive)