91Ó°ÊÓ

\(\sqrt[3]{r^{2}+2 r+8}=\sqrt[3]{r^{2}+3 r+12}\)

Consider the expression $$ \sqrt{63}+\sqrt{112}-\sqrt{252} $$ (a) Simplify this expression using the methods of this section. (b) Use a calculator to approximate the given expression. (c) Use a calculator to approximate the simplified expression in part (a). (d) Complete the following: Assuming the work in part (a) is correct, the approximations in parts (b) and (c) should be ( equal / unequal).

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ \frac{5 \sqrt{2 m}}{\sqrt{y^{3}}} $$

Simplify each root. $$ -\sqrt[6]{(-5)^{6}} $$

Simplify each expression. Assume that all variables represent positive real numbers. $$ \frac{64^{5 / 3}}{64^{4 / 3}} $$

\(\sqrt[3]{x^{2}+7 x+2}=\sqrt[3]{x^{2}+6 x+1}\)

Let \(a=1\) and let \(b=64\). (a) Evaluate \(\sqrt{a}+\sqrt{b}\). Then find \(\sqrt{a+b}\). Are they equal? (b) Evaluate \(\sqrt[3]{a}+\sqrt[3]{b}\). Then find \(\sqrt[3]{a+b}\). Are they equal? (c) Complete the following: In general, $$ \sqrt[n]{a}+\sqrt[n]{b} \neq $$ based on the observations in parts (a) and (b) of this exercise.

Simplify each expression. Assume that all variables represent positive real numbers. $$ \frac{125^{7 / 3}}{125^{5 / 3}} $$

Simplify. \(\sqrt[5]{128}\)

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ \frac{2 \sqrt{5 r}}{\sqrt{m^{3}}} $$