Writing Radical Expressions as Exponential Expressions
\[ \Large \sqrt[n]{a^m } = a^{\frac{m}{n} } \]
Example:
\(\bullet \large\sqrt[4]{ 8} = \sqrt[4]{2^3 } = 2^{ \frac{3}{4} } \)
\(\bullet \large\sqrt[3]{ -2} = (-2)^{ \frac{1}{3} } \)
Question 2
If \[ \large \sqrt[3]{2^x } = \sqrt{(0.5)^{2x-1} } \]
find x.
\[ \text{A)} \ 3 \quad \text{B) } \ 2 \quad \text{C) } \ 1 \quad \text{D) } \ \frac{3}{8} \quad \text{E)} \ \frac{1}{6} \]
Solution:
\[ \large\sqrt[3]{2^x} = \sqrt{(0.5)^{2x-1}} \]
\[ \large \Rightarrow 2^{\Large\frac{x}{3}} = \left( \frac{1}{2} \right)^{\Large\frac{2x-1}{2}} \]
\[ \large \Rightarrow 2^{\Large\frac{x}{3}} = (2^{-1})^{\Large\frac{2x-1}{2}} \]
\[ \large \Rightarrow 2^{\Large\frac{x}{3}} = 2^{\Large\frac{-2x+1}{2}} \]
\[ \Rightarrow \frac{x}{3} = \frac{1 – 2x}{2} \] \[ \Rightarrow x = \frac{3}{8} \]
\(\textbf{Answer: D} \)
Question 3
If
\[ \large\sqrt[4]{\left( \frac{3}{20} \right)^{3x+1}} = \frac{400}{9} \]
find x.
\[ \text{A)} \ -4 \quad \text{B) } \ -3 \quad \text{C) } \ 0 \quad \text{D) } \ 3 \quad \text{E)} \ 4 \]
Solution:
\[ \sqrt[4]{\left( \frac{3}{20} \right)^{3x+1}} = \frac{400}{9} \]
\[ \left( \frac{3}{20} \right)^{\large\frac{3x+1}{4}} = \left( \frac{20}{3}\right)^2 = \left( \frac{3}{20}\right)^{-2} \]
\[ \Rightarrow \frac{3x +1 }{4} = -2 \]
\[ \Rightarrow 3x +1 = -8 \]
\[ \Rightarrow 3x = -9 \]
\[ \Rightarrow x= -3 \]
\(\textbf{Answer: B} \)
← Previous Page | Next Page →