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Chapter 11: Problem 8

Quantity \(\mathbf{A}\) $$\frac{2^{-4}}{4^{-2}}$$ Quantity_B $$\frac{\sqrt{64}}{-2^{3}}$$ a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.

Short Answer

Expert verified
a. Quantity A is greater.

Step by step solution

01

Simplify Quantity A

To simplify the expression \(\frac{2^{-4}}{4^{-2}}\), we first examine the exponents. The base number 2 has an exponent of -4 and the base number 4 has an exponent of -2. A negative exponent can be converted to a positive exponent by taking the reciprocal of the base number. So, \(2^{-4}\) is the same as \(\frac{1}{2^4}\) and \(4^{-2}\) is equivalent to \(\frac{1}{4^2}\). Thus, the original expression simplifies to \(\frac{\frac{1}{2^4}}{\frac{1}{4^2}}\). By multiplying the numerator and denominator by \(2^4\) and \(4^2\) respectively, the expression simplifies to \( \frac{4^2}{2^4} \). Calculating \(4^2\) gives 16 and calculating \(2^4\) gives 16 as well. Hence, the simplified form of the expression is 1.
02

Simplify Quantity B

To simplify the expression \(\frac{\sqrt{64}}{-2^{3}}\), we first compute the square root of 64, which is 8. Then we compute \( -2^3 \), which is -8. So, the expression simplifies to \(\frac{8}{-8}\). Thus, the simplified form of the expression is -1.
03

Comparing Quantities A and B

Now that we have simplified forms of both quantities, we can easily compare them. Since 1 is greater than -1, it can be concluded that Quantity A is greater than Quantity B.

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