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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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Most popular questions from this chapter

$$x^{2}+8 x=-7$$ Quantity \(\mathbf{A}\) \(x\) Quantity_B 0 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.

If \(A=2 x-(y-2 c)\) and \(B=(2 x-y)-2 c,\) then \(A-B=\) a. \(-2 y\) b. \(-4 c\) c. 0 d. \(2 y\) e. \(4 c\)

If \(a\) is multiplied by 3 and the result is 4 less than 6 times \(b,\) what is the value of \(a-2 b ?\) a. -12 b. \(-\frac{4}{3}\) c. \(-\frac{3}{4}\) d. \(\frac{4}{3}\) e. 12

If \(x=3 a\) and \(y=9 b,\) then all of the following are equal to \(2(x+y)\) EXCEPT a. \(3(2 a+6 b)\) b. \(6(a+3 b)\) c. \(24\left(\frac{1}{4} a+\frac{3}{4} b\right)\) d. \(\frac{1}{3}(18 a+54 b)\) e. \(12\left(\frac{1}{2} a+\frac{3}{4} b\right)\)

A merchant sells three different sizes of canned tomatoes. A large can costs the same as 5 medium cans or 7 small cans. If a customer purchases an equal number of small and large cans of tomatoes for the same amount of money needed to buy 200 medium cans, how many small cans does she purchase? a. 35 b. 45 c. 72 d. 99 e. 208
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