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Chapter 0: Problem 41

Simplify each exponential expression. $$\left(-\frac{4}{x}\right)^{3}$$

Short Answer

Expert verified
The simplified form of the given exponential expression is \(-64/x^3\).

Step by step solution

01

Analyze the expression

In the given notation \(\left(-\frac{4}{x}\right)^{3}\), the main operation is the exponentiation. The exponential operation is denoted with the symbol ^, and the number 3 is the exponent. In this exercise, the exponent is a positive integer, so we simply need to replicate \(-\frac{4}{x}\) three times and multiply them.
02

Apply exponentiation

Since the exponent is 3, which is an odd number, the negative sign remains due to the property of negative numbers raised to odd powers. We replicate \(-\frac{4}{x}\) three times, multiplying them together: \(-\frac{4}{x} * -\frac{4}{x} * -\frac{4}{x}\).
03

Simplify the expression

We simplify the expression: \(-4*-4*-4 = -64\) and \(\frac{1}{x} * \frac{1}{x} * \frac{1}{x} = \frac{1}{x^3}\), so the simplified expression is \(-64/x^3\).

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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 all integers \(b\) so that the trinomial can be factored. $$x^{2}+b x+15$$

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

$$\frac{5}{4} \cdot \frac{8}{15}$$

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