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Chapter 0: Problem 109

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(2 x^{-3} y z^{-6}\right)(2 x)^{-5}$$

Short Answer

Expert verified
The simplified form of the given expression is \(y/16x^{8}z^{6}\).

Step by step solution

01

- Identify the base with its exponent

Here we first examine the original expression \(\left(2 x^{-3} y z^{-6}\right)(2 x)^{-5}\). In the first bracket, 2, \(x^{-3}\), y, and \(z^{-6}\) are all separate bases, each to the power of 1, -3, 1, and -6 respectively. In the second bracket, (2x) is the base and -5 is the exponent.
02

- Distribute exponent in the second bracket

Next, distribute the exponent to both parts of the base. This is done by multiplying -5 to the exponents of the base. We get \(2^{-5} x^{-5}\).
03

- Combine like bases

Then we combine the like bases by adding the exponents: two for 2 and two for x. Hence, we have \(2^{1-5} x^{-3 -5} y z^{-6}\). So far, we obtain: \(2^{-4} x^{-8} y z^{-6}\).
04

- Interpret negative exponents

Note that any non-zero base with negative exponent implies a reciprocal of the base with a positive exponent. Thus, \(2^{-4}\) becomes \(1/2^{4}\), \(x^{-8}\) becomes \(1/x^{8}\), and \(z^{-6}\) becomes \(1/z^{6}\). Rewriting, we get: \(1/2^{4} \cdot 1/x^{8} \cdot y \cdot 1/z^{6}\).
05

- Simplify

The last step is to simplify: \(1/2^{4}\) equals 1/16. So, we have \(1/16 x^{-8} y z^{-6}\) considering multiplication in mathematics is associative. It can be written as \(y/16x^{8}z^{6}\).

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Most popular questions from this chapter

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Find the intersection of the sets. $$\\{1,3,7\\} \cap(2,3,8)$$

$$\text { factor completely.}$$ $$-x^{2}-4 x+5$$

$$\text { Factor completely.}$$ $$2 x^{2}-7 x y^{2}+3 y^{4}$$

$$\text { Factor completely.}$$ $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$
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