Walkabout wrote:
\((\frac{1}{2})^{-3}(\frac{1}{4})^{-2}(\frac{1}{16})^{-1}=\)
(A) (1/2)^(-48)
(B) (1/2)^(-11)
(C) (1/2)^(-6)
(D) (1/8)^(-11)
(E) (1/8)^(-6)
We start by using the negative exponent rule. When a fractional base is raised to a negative exponent, we can rewrite the expression (without the negative exponent) by flipping the fraction and making the exponent positive. For example, (1/2)^-3 = 2^3
We are using the negative exponent rule because it’s not only easier to deal with positive exponents, but also when we flip the fractional base, the fraction becomes an integer.
(1/2)^-3 = 2^3
(1/4)^-2 = 4^2 = (2 x 2)^2 = (2^2)^2 = 2^4
(1/16)^-1 = 16^1 = (2 x 2 x 2 x 2)^1 = (2^4)^1 = 2^4
We multiply each term in the expression, obtaining:
2^3 x 2^4 x 2^4
Remember, since the bases are the same, we keep the base and add the exponents. We are left with:
2^(3+4+4) = 2^11
Finally, since our answer choices are expressed in fractional form, we once again have to use the negative exponent rule. To convert a base with a positive exponent, take the reciprocal of the base and change the positive exponent to its negative counterpart. Using the rule we get:
2^11 = (1/2)^-11
The answer is B.
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