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Express the given expression without logs, in simplest form. Assume all variables represent positive values.

(8^(-(1)/(3)log_(8)27w^(3)))
Answer:

Express the given expression without logs, in simplest form. Assume all variables represent positive values. $\newline$ $\left(8^{-\frac{1}{3} \log _{8} 27 w^{3}}\right)$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

Q. Express the given expression without logs, in simplest form. Assume all variables represent positive values.

\newline

\left(8^{-\frac{1}{3} \log _{8} 27 w^{3}}\right)

\newline

Answer:

Rewrite Expression: Understand the given expression and rewrite it for clarity. $\newline$ The given expression is $8^{-\left(\frac{1}{3}\log_{8}27w^{3}\right)}$ . We need to express this without logs.
Apply Power Rule: Apply the power rule of logarithms. $\newline$ The power rule states that $a^{\log_a(b)} = b$ . In this case, we have a negative exponent and a fraction, so we need to apply the rule carefully.
Rewrite with Power Rule: Rewrite the expression using the power rule. $\newline$ The expression $8^{-\left(\frac{1}{3}\right)\log_{8}27w^{3}}$ can be rewritten as $\left(8^{\log_{8}27w^{3}}\right)^{-\left(\frac{1}{3}\right)}$ .
Simplify Inside Parentheses: Simplify the expression inside the parentheses using the power rule. $\newline$ Since $8^{\log_{8}27w^{3}} = 27w^{3}$ , we can replace the expression inside the parentheses with $27w^{3}$ .
Apply Negative Exponent Rule: Apply the negative exponent rule. $\newline$ The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . Therefore, $(27w^{3})^{-\left(\frac{1}{3}\right)}$ becomes $\frac{1}{\left(27w^{3}\right)^{\frac{1}{3}}}$ .
Simplify Inside Parentheses: Simplify the expression inside the parentheses using the cube root. $\newline$ The cube root of $27w^{3}$ is $3w$ because $(3w)^3 = 27w^{3}$ . So, we have $1/((3w)^3)^{(1/3)}$ .
Simplify with Exponents: Simplify the expression using the property of exponents. $\newline$ Since $((3w)^3)^{\frac{1}{3}}$ is the same as $(3w)$ , the expression simplifies to $\frac{1}{3w}$ .

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