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$(4^{-2} \times 4^{-3})^{3}$

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Understanding negative exponents

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Q.

(4^{-2} \times 4^{-3})^{3}

Identify Base and Exponents: Identify the base and the exponents in the expression $(4^{-2} \times 4^{-3})^{3}$ . We have the base $4$ raised to the exponents $-2$ and $-3$ , which are then raised to the power of $3$ .
Use Exponent Property: Use the property of exponents that states when multiplying like bases, you add the exponents. $\newline$ Combine the exponents of $4$ by adding them together: $4^{-2} \times 4^{-3} = 4^{(-2 + -3)}$ . $\newline$ Calculate the sum of the exponents: $-2 + -3 = -5$ . $\newline$ So, $4^{-2} \times 4^{-3} = 4^{-5}$ .
Apply Power Rule: Apply the power of a power rule, which states $(a^b)^c = a^{(b*c)}$ . $\newline$ Raise the result of Step $2$ to the power of $3$ : $(4^{-5})^3$ . $\newline$ Calculate the new exponent: $-5 * 3 = -15$ . $\newline$ So, $(4^{-5})^3 = 4^{-15}$ .
Convert to Fraction: Convert the expression with a negative exponent to a fraction. $4^{-15}$ is the same as $1 / 4^{15}$ .

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