Question

select all the correct answers.
which of the following properties can be used to show that the expression $5^{\frac{3}{2}}$ is equivalent to $\sqrt{5^3}$?
$\left(5^{\frac{3}{2}}\
ight)^2 = 5^{(\frac{3}{2} \cdot 2)} = 5^3$
$\frac{5^{\frac{7}{2}}}{5^{\frac{1}{2}}} = 5^{(\frac{7}{2} - \frac{1}{2})} = 5^3$
$\sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{\frac{3}{2}}$
$(5^9)^{\frac{1}{3}} = 5^{(9 \cdot \frac{1}{3})} = 5^3$
$5^{\frac{5}{2}} \cdot 5^{\frac{1}{2}} = 5^{(\frac{5}{2} + \frac{1}{2})} = 5^3$

Explanation:

Step1: Identify root-exponent equivalence

Recall that $\sqrt{x} = x^{\frac{1}{2}}$, so $\sqrt{5^3} = (5^3)^{\frac{1}{2}}$.

Step2: Verify power of a power rule

For $(5^{\frac{3}{2}})^2$, use $(a^m)^n = a^{m \cdot n}$:
$(5^{\frac{3}{2}})^2 = 5^{\frac{3}{2} \cdot 2} = 5^3$. Squaring both $5^{\frac{3}{2}}$ and $\sqrt{5^3}$ gives $5^3$, proving they are equivalent.

Step3: Verify reverse root to exponent

For $\sqrt{5^3} = (5^3)^{\frac{1}{2}}$, use $(a^m)^n = a^{m \cdot n}$:
$(5^3)^{\frac{1}{2}} = 5^{3 \cdot \frac{1}{2}} = 5^{\frac{3}{2}}$, directly showing equivalence.

Step4: Eliminate unrelated options

Options 2,4,5 manipulate exponents to get $5^3$ but do not connect $5^{\frac{3}{2}}$ and $\sqrt{5^3}$.

Answer:

$\boldsymbol{(5^{\frac{3}{2}})^2 = 5^{(\frac{3}{2} \cdot 2)} = 5^3}$
$\boldsymbol{\sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{\frac{3}{2}}}$