select all the correct answers.\nwhich of the following properties can be used to show that the expression…

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

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

Answer

Explanation:

Step1: Recall rational exponent rule

The rule states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, and $(a^m)^n = a^{m \cdot n}$.

Step2: Analyze first option

This squares both expressions, but does not directly show $5^\frac{3}{2} = \sqrt{5^3}$, it verifies equality after squaring, not the equivalence of the original expressions.

Step3: Analyze second option

This uses quotient of powers, which simplifies to $5^3$, not connecting $5^\frac{3}{2}$ and $\sqrt{5^3}$.

Step4: Analyze third option

Applies the rational exponent rule directly: rewrite radical as exponent, $\sqrt{5^3} = (5^3)^\frac{1}{2} = 5^{\frac{3}{2}}$, which shows the equivalence.

Step5: Analyze fourth option

This simplifies to $5^3$, unrelated to the target equivalence.

Step6: Analyze fifth option

This uses product of powers to get $5^3$, not connecting the two target expressions.

Answer:

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