91Ó°ÊÓ

Simplify \(\sqrt[5]{64}\)

Solve each equation. $$ \sqrt[3]{r^{2}+2 r+8}=\sqrt[3]{r^{2}+3 r+12} $$

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).

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

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

\(\sqrt{1500}\)

Solve each equation. $$ \sqrt[3]{x^{2}+7 x+2}=\sqrt[3]{x^{2}+6 x+1} $$

Concept Check Without using a calculator, determine between which two consecutive integers each square root lies. For example, \(\sqrt{75}\) is between 8 and \(9,\) because \(\sqrt{64}=8, \sqrt{81}=9,\) and \(64<75<81\). \(\sqrt{94}\)

Simplify each expression. Assume that all variables represent positive real numbers. $$ y^{7 / 3} \cdot y^{-4 / 3} $$