Exponents and Radicals

\(\begin{aligned}&\text{Which expression is equivalent to }(m^4q^4z^{-1})(mq^5z^3),\text{where }m,q,\mathrm{and~}z\mathrm{~are~positive?}\end{aligned}\)

\(\begin{aligned}&\text{Which expression is equivalent to }\frac{6v^{12}}{18v^3},\text{ where }v>0?\end{aligned}\)

\(\begin{aligned}&\text{Which expression is equivalent to }\frac{h^{15}q^{7}}{h^{5}q^{21}},\text{where }h>0\text{ and }q>0?\end{aligned}\)

\(\begin{aligned}
&\text{The expression }\frac{x^{20}(x-4)}{5x^{2}}+\frac{4x^{20}}{5x^{2}}\text{ is equivalent to }\frac{1}{5}x^{c},\text{where }c\text{ is a constant and }x>0.\text{ What }\\
&\text{is the value of }c?\end{aligned}\)

\(\begin{aligned}&\text{If }3x-y=12,\text{ what is the value of }\frac{8^{x}}{2^{y}}?\end{aligned}\)

\(\begin{aligned}&\mathrm{If~}\frac{x^{a^{2}}}{x^{b^{2}}}=x^{16},x>1,\mathrm{and~}a+b=2,\text{what is the value of }a-b ?\end{aligned}\)

\(\begin{aligned}&\text{Which of the following is equal to }a^\frac{2}{3},\text{for all values of }a?\end{aligned}\)

\(\text{Which expression is equivalent to }\sqrt{x^{197}},\text{where }x > 0?\)

\(\begin{aligned}&
\text{The expression }\frac{x^{-2}y^{\frac{1}{2}}}{x^{\frac{1}{3}}y^{-1}},\text{where }x>1\text{ and }y>1,\text{ is equivalent to which of the following?}\end{aligned}\)

\(\begin{aligned}&\text{Which expression is equivalent to }a^{\frac{11}{12}},\text{ where }a > 0?\end{aligned}\)

\(\begin{aligned}\text{Which expressions is equivalent to }(131y)^{\frac{1}{2}},\text{ where }y>1?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{a^{2}b^{6}}\\
&\text{If }a\text{ and }b\text{ are positive numbers, which of the following is equivalent to the }\\&\text{expression above?}\end{aligned}\)

\(\begin{aligned}&\text{Which expression is equivalent to }\sqrt[12]{x^{5}y^{5}},\text{ where }x\text{ and }y\text{ are positive?}\end{aligned}\)

\(\begin{aligned}&\text{Which expression is equivalent to }\sqrt[5]{32a^3b^4},\text{ where }a>0\text{ and }b >0?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{\sqrt[3]{x^{17}y^{5}}}{5x^{2}\left(\sqrt[8]{y^{8}}\right)}\\&\text{For all positive values of }x\mathrm{~and~}y,\text{ the given expression is equivalent to which of the}\\&\text{following?}\\
&~~~~~~~~~\text{I. }~~\frac{\left(x^{\frac{14}{3}}\right)\left(y^{\frac{5}{3}}\right)}{5xy}\\
&~~~~~~~~\text{II. }~~\frac{\sqrt[3]{x^{11}y^{2}}}{5}\end{aligned}\)

\(\text{If }4^{8c}=\sqrt[3]{4^7}~,\text{ what is the value of }c?\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt[5]{x^{m}}\\&\text{In the given expression, }m\text{ is a constant and }x > 1.\text{ The expression can be rewritten as }\sqrt[7]{x}.\\
&\text{What is the value of }m?\end{aligned}\)

\(\begin{aligned}
&\text{The expression }6\sqrt[5]{3^{5}x^{45}}\cdot\sqrt[8]{2^{8}x}\text{ is equivalent to }ax^{b}\text{ where }a\text{ and }b\text{ are positive constants }\\&\text{and }x>1.\text{ What is the value of }a+b?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt[5]{119n}\left(\sqrt[6]{119n}\right)^2\\&\text{For what value of }x\text{ is the given expression equivalent to }(119n)^{30x},\text{ where }n > 1?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt[5]{p^{2}}=t^{\frac{3}{4}}\\&\text{In the given equation, }p > 1\text{ and }t>1.\text{ If }t=p^{2n-1}\text{ where }n\text{ is a constant, what is the value}\\&\text{of }n?\end{aligned}\)

\(\begin{aligned}&\text{If }n\text{ and }k\text{ are numbers greater than }1\text{ and }\sqrt[4]{n^5}\text{ is equivalent to }\sqrt[3]{k^2},\text{ for what value of }a\text{ is}\\&n^{2a+1}\text{ equal to }k?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{x^{2}}=x\\&\text{Which of the following values of }x\text{ is NOT a solution to the equation above?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{(x+15)^2}=x+15\\
&\text{What is the least value of }x\text{ that is a solution to the given equation?}\end{aligned}\)

\(\begin{aligned}&\mathrm{If~}\sqrt{x}+\sqrt{9}=\sqrt{64},\text{what is the value of }x?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{2x+19}=5\\
&\text{What is the solution to the equation shown?}\end{aligned}\)

\(\text{What is the solution to the equation }2\sqrt{x}=3?\)

\(\text{If }4\sqrt{2x}=16,\text{what is the value of }6x?\)

\(\text{If }a=5\sqrt{2}\text{ and }2a=\sqrt{2x},\text{ what is the value of }x?\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{k+2}-x=0\\&\text{In the equation above, }k\text{ is a constant. If }x=9,\text{ what is the value of }k ?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f(x)=\sqrt{5x+6}\\&\text{The function }f\text{ is defined by the given equation. If }f(a)=-5a,\text{ where }a\text{ is a constant,}\\&\text{what is the value of }a?\\\end{aligned}\)

\(\text{What is the set of all solutions to the equation }\sqrt{x+2}=-x?
\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~\sqrt{2x+6}+4=x+3\\&\text{What is the solution set of the equation above?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{x-a}=x-4\\&\mathrm{If~}a=2,\text{what is the solution set of the equation above?}
\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{4x}=x-3\\&\text{What are all values of }x\text{ that satisfy the given equation?}\\&~~~~\text{I. 1}\\&~~~~\text{II. 9}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\sqrt{x^2-15x+175}=x\sqrt{11}\\
&\text{What are all possible solutions to the given equation?}\end{aligned}\)

\(\begin{aligned}
&\text{If }x=\sqrt[{2n}]{2x^{n}+35},\text{ where }n\text{ is a positive integer constant, what is the value }\\
&\text{of }x^{n}~?\end{aligned}\)