Simplify the Radical Expression: (⁶√5³)/(³√5³) Step by Step

To solve this problem, we'll follow these steps:

• Step 1: Convert the given roots into fractional exponents.

• Step 2: Simplify the fractional exponents where necessary.

• Step 3: Divide using the properties of exponents.

• Step 4: Simplify the resulting expression.

Now, let's work through each step:

Step 1: Convert the roots to fractional exponents.
The expression $\frac{\sqrt[6]{5^3}}{\sqrt[3]{5^3}}$ becomes $\frac{(5^3)^{\frac{1}{6}}}{(5^3)^{\frac{1}{3}}}$.

Step 2: Simplify each fractional exponent.
We know $(a^m)^n = a^{m \cdot n}$. So, apply this rule: -

$(5^3)^{\frac{1}{6}} = 5^{3 \cdot \frac{1}{6}} = 5^{\frac{3}{6}} = 5^{\frac{1}{2}}$.

$(5^3)^{\frac{1}{3}} = 5^{3 \cdot \frac{1}{3}} = 5^{\frac{3}{3}} = 5^1 = 5$.

Step 3: Divide using the properties of exponents.
The division of powers with the same base: $a^m / a^n = a^{m-n}$. Thus, $\frac{5^{\frac{1}{2}}}{5}$ simplifies to $5^{\frac{1}{2} - 1}$.

Step 4: Simplify the resulting exponent expression.
$5^{\frac{1}{2} - 1} = 5^{-\frac{1}{2}} = \frac{1}{5^{\frac{1}{2}}} = \sqrt{5}$, (since the negative exponent indicates reciprocal).

Therefore, the solution to the problem is $\sqrt{5}$, which matches Choice 2.