Question

simplify the expression using the properties of exponents.\\(\frac{(2^3)^4 cdot 3^2 cdot 3^3}{2^5 cdot (2 cdot 3)^3}\\)\\(\circledcirc\\) apply the power of a power property to \\((2^3)^4\\) in the numerator.

Explanation:

Step1: Power of a power rule

$(2^3)^4 = 2^{3 \times 4} = 2^{12}$

Step2: Combine like bases (numerator)

$2^{12} \cdot 3^2 \cdot 3^5 = 2^{12} \cdot 3^{2+5} = 2^{12} \cdot 3^7$

Step3: Expand denominator product

$(2 \cdot 3)^5 = 2^5 \cdot 3^5$, so denominator is $2^5 \cdot 2^5 \cdot 3^5 = 2^{5+5} \cdot 3^5 = 2^{10} \cdot 3^5$

Step4: Subtract exponents (like bases)

For base 2: $2^{12-10}=2^2$
For base 3: $3^{7-5}=3^2$

Step5: Compute final simplified form

$2^2 \cdot 3^2 = (2 \cdot 3)^2 = 6^2$

Answer:

$2^2 \cdot 3^2$ or $6^2$ (or $36$ if evaluated, but per exponent property focus, $2^2 \cdot 3^2$ is the simplified exponential form)