Solve the Nested Expression: ((b×6)^5)^2 Using Exponent Rules

To solve the problem, we need to simplify the expression $\left(\left(b \times 6\right)^5\right)^2$.

We will apply the power of a power rule in exponents, which states:

• For an expression $(x^m)^n$, it simplifies to $x^{m \times n}$.

Applying this rule to our expression:

$\left(\left(b \times 6\right)^5\right)^2 = \left(b \times 6\right)^{5 \times 2}$

Calculating the new exponent:

$5 \times 2 = 10$

Therefore, the simplified expression is:

$\left(b \times 6\right)^{10}$

We will now compare this to the given multiple-choice answers:

• Choice 1: $\left(b\times6\right)^3$ - Incorrect
• Choice 2: $\left(b\times6\right)^{10}$ - Correct
• Choice 3: $\left(b\times6\right)^7$ - Incorrect
• Choice 4: $\left(b\times6\right)^{\frac{2}{5}}$ - Incorrect

In conclusion, the correct answer is $\left(b\times6\right)^{10}$, which matches Choice 2.