Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

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

• Step 1: Identify the common exponent in the expression.
• Step 2: Use the power of a product rule to rewrite the expression with a single exponent.

Now, let's work through each step:

Step 1: Recognize that each term in the expression $5^{2a} \times 8^{2a} \times 7^{2a}$ has the same exponent, which is $2a$.

Step 2: Apply the power of a product rule. This rule states that $(x \times y \times z)^n = x^n \cdot y^n \cdot z^n$, which can be reversed to combine the terms.

With this understanding, the expression $5^{2a} \times 8^{2a} \times 7^{2a}$ can be rewritten as a single exponent expression by combining the bases:

$(5 \cdot 8 \cdot 7)^{2a}$.

Therefore, the solution to this problem is $\left(5 \times 8 \times 7\right)^{2a}$.