# Explanation: ## Step1: Analyze the expression The given expression is $81x^{20}y^{24}z^{8}$. We can try to express each part as a power of some base. - For the coefficient: $81 = 3^4$ (since $3\times3\times3\times3 = 81$). - For the $x$ term: $x^{20}=(x^{5})^4$ (using the power - of - a - power rule $(a^m)^n=a^{mn}$, here $m = 5$, $n = 4$ and $5\times4=20$). - For the $y$ term: $y^{24}=(y^{6})^4$ (since $6\times4 = 24$). - For the $z$ term: $z^{8}=(z^{2})^4$ (since $2\times4=8$). ## Step2: Apply the product of powers rule The product of powers rule states that $a^m\times b^m=(ab)^m$. So we can rewrite the expression as: \[ \begin{align*} 81x^{20}y^{24}z^{8}&=3^{4}\times(x^{5})^{4}\times(y^{6})^{4}\times(z^{2})^{4}\\ &=(3x^{5}y^{6}z^{2})^{4} \end{align*} \] # Answer: $(3x^{5}y^{6}z^{2})^{4}$

81x^{20}y^{24}z^{8}

QUESTION IMAGE

$81x^{20}y^{24}z^{8}$

Question

81x^{20}y^{24}z^{8}

Explanation:

Step1: Analyze the expression

The given expression is $81x^{20}y^{24}z^{8}$. We can try to express each part as a power of some base.

For the coefficient: $81 = 3^4$ (since $3\times3\times3\times3 = 81$).
For the $x$ term: $x^{20}=(x^{5})^4$ (using the power - of - a - power rule $(a^m)^n=a^{mn}$, here $m = 5$, $n = 4$ and $5\times4=20$).
For the $y$ term: $y^{24}=(y^{6})^4$ (since $6\times4 = 24$).
For the $z$ term: $z^{8}=(z^{2})^4$ (since $2\times4=8$).

Step2: Apply the product of powers rule

The product of powers rule states that $a^m\times b^m=(ab)^m$. So we can rewrite the expression as:

$$ LATEXBLOCK0 $$

Answer:

$(3x^{5}y^{6}z^{2})^{4}$