Simplify the Exponential Expression: (3×7)^(2x+5) ÷ (3×7)^(x+3)

To solve the problem $\frac{\left(3\times7\right)^{2x+5}}{\left(3\times7\right)^{x+3}}=$ , we need to apply the Power of a Quotient Rule for Exponents.

The Power of a Quotient Rule states that $\frac{a^m}{a^n} = a^{m-n}$ where $a$ is a nonzero number and $m$ and $n$ are integers. In this expression, $a$ will be equal to $(3 \times 7)$.

Start by writing the expression in a simplified form using the rule:

• The numerator is $\left(3 \times 7\right)^{2x + 5}$
• The denominator is $\left(3 \times 7\right)^{x + 3}$

Applying the quotient rule:

$\frac{\left(3 \times 7\right)^{2x+5}}{\left(3 \times 7\right)^{x+3}} = \left(3 \times 7\right)^{(2x+5) - (x+3)}$

Now we simplify the exponent:

• $(2x + 5) - (x + 3) = 2x + 5 - x - 3$
• Combine like terms: $2x - x + 5 - 3 = x + 2$

Thus, $\frac{\left(3 \times 7\right)^{2x+5}}{\left(3 \times 7\right)^{x+3}} = \left(3 \times 7\right)^{x+2}$.

The solution to the question is: $\left(3 \times 7\right)^{x+2}$