Which expressions are equivalent to $\newline$ $4^{-2} \cdot 7^{-2}$ ? $\newline$ Choose $2$ answers: $\newline$ A) $\newline$ $(4 \cdot 7)^{-4}$ $\newline$ B) $\newline$ $\frac{1}{28^{2}}$ $\newline$ C) $\newline$ $\frac{7^{-2}}{4^{2}}$ $\newline$ D) $\newline$ $(4 \cdot 7)^{4}$

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Math Problems

Grade 8

Evaluate expressions using properties of exponents

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Q. Which expressions are equivalent to

\newline

4^{-2} \cdot 7^{-2}

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Choose

2

answers:

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(4 \cdot 7)^{-4}

\newline

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\frac{1}{28^{2}}

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\frac{7^{-2}}{4^{2}}

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(4 \cdot 7)^{4}

Understanding the expression: Understand the given expression and the properties of exponents. $\newline$ The given expression is $4^{-2} \times 7^{-2}$ . According to the properties of exponents, when multiplying powers with the same exponent but different bases, you can combine the bases and keep the exponent.
Combining the bases: Apply the property of exponents to combine the bases. $\newline$ $4^{-2} \cdot 7^{-2}$ can be written as $(4 \cdot 7)^{-2}$ because the exponents are the same. $\newline$ Calculation: $(4 \cdot 7)^{-2} = 28^{-2}$
Comparing the choices: Compare the result with the given choices. $\newline$ We have $(4 \times 7)^{-2}$ which is equivalent to $28^{-2}$ . Now we need to find which choices match this expression.
Analyzing choice A: Analyze choice A. $\newline$ Choice A is $(4 \times 7)^{-4}$ . This is not equivalent to $28^{-2}$ because the exponent is $-4$ , not $-2$ .
Analyzing choice B: Analyze choice B. $\newline$ Choice B is $(1)/(28^{2})$ . This is equivalent to $28^{-2}$ because a negative exponent indicates the reciprocal of the base raised to the positive exponent. $\newline$ Calculation: $28^{-2} = 1/(28^2)$
Analyzing choice C: Analyze choice C. $\newline$ Choice C is $(7^{-2})/(4^{2})$ . This is not equivalent to $28^{-2}$ because the bases are not multiplied together, and the exponents are not the same.
Analyzing choice D: Analyze choice D. $\newline$ Choice D is $(4 \times 7)^{4}$ . This is not equivalent to $28^{-2}$ because the exponent is $4$ , not $-2$ , and it is positive, not negative.
Concluding the correct choices: Conclude with the correct choices. $\newline$ The correct choices that are equivalent to $4^{-2}\times 7^{-2}$ are B) $\left(\frac{1}{28^{2}}\right)$ and none of the other choices match.