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Simplify the following expression to simplest form using only positive exponents.

(64x^(-3)y^(-15))^((4)/(3))
Answer:

Simplify the following expression to simplest form using only positive exponents. $\newline$ $\left(64 x^{-3} y^{-15}\right)^{\frac{4}{3}}$ $\newline$ Answer: $\newline$

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Multiplication with rational exponents

Full solution

Q. Simplify the following expression to simplest form using only positive exponents.

\newline

\left(64 x^{-3} y^{-15}\right)^{\frac{4}{3}}

\newline

Answer:

\newline

Apply Power to Factors: Apply the power to each factor inside the parentheses. $\newline$ When raising a power to a power, you multiply the exponents. In this case, we have a fractional exponent $(4/3)$ , which we will apply to each factor inside the parentheses. $\newline$ $(64x^{-3}y^{-15})^{(4)/(3)} = 64^{(4)/(3)} * x^{-3*(4)/(3)} * y^{-15*(4)/(3)}$
Simplify Each Factor: Simplify each factor separately. $\newline$ First, we simplify $64^{(4)/(3)}$ . Since $64$ is $4^3$ , we can rewrite this as $(4^3)^{(4)/(3)}$ . $\newline$ $64^{(4)/(3)} = (4^3)^{(4)/(3)} = 4^{3*(4)/(3)} = 4^4 = 256$ $\newline$ Next, we simplify the $x$ and $y$ terms by multiplying the exponents. $\newline$ $x^{(-3*(4)/(3))} = x^{-4}$ $\newline$ $y^{(-15*(4)/(3))} = y^{-20}$
Rewrite with Positive Exponents: Rewrite the expression with positive exponents. $\newline$ To convert negative exponents to positive, we take the reciprocal of the base raised to the positive of the negative exponent. $\newline$ $x^{-4}$ becomes $\frac{1}{x^4}$ $\newline$ $y^{-20}$ becomes $\frac{1}{y^{20}}$
Combine Simplified Factors: Combine the simplified factors. $\newline$ Now we combine the simplified factors to get the final expression. $\newline$ $256 \times \left(\frac{1}{x^4}\right) \times \left(\frac{1}{y^{20}}\right)$
Write Final Answer: Write the final answer. $\newline$ The final simplified expression with positive exponents is: $\newline$ $\frac{256}{x^4y^{20}}$

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