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Chapter 3: Problem 38

Rewrite each expression as simply as you can. $$\left(x^{-2}\right)^{3} \cdot x^{5}$$

Short Answer

Expert verified
\( \frac{1}{x} \)

Step by step solution

01

Apply the Power Rule

First, apply the power rule \( \left(a^m\right)^n = a^{m \cdot n} \) to the expression \( \left(x^{-2}\right)^3 \). This simplifies to \[ x^{-2 \cdot 3} = x^{-6} \]
02

Combine Like Bases

Now multiply \( x^{-6} \) by \( x^5 \) by adding the exponents because the bases are the same: \[ x^{-6} \cdot x^5 = x^{-6+5} = x^{-1} \]
03

Simplify the Exponent

Finally, simplify \( x^{-1} \). Recall that \( x^{-1} = \frac{1}{x} \), so the simplified expression is \[ \frac{1}{x} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule in algebra is a handy tool for simplifying expressions with exponents. When you encounter an expression such as \( (a^m)^n \), the power rule states that you can multiply the exponents. This gives us \[ a^{m \times n} \].

For example, in the expression \( (x^{-2})^3 \), we apply the power rule. We multiply the exponents -2 and 3, resulting in \[ x^{-2 \times 3} = x^{-6} \].
This step helps us to simplify complex exponent terms into a more manageable form.
Negative Exponents
Negative exponents might seem confusing at first, but they simplify expressions in useful ways. A negative exponent indicates that the base should be taken as the reciprocal. In mathematical terms, \[ x^{-n} = \frac{1}{x^n} \].

So, if you see an expression like \( x^{-6} \), it鈥檚 equivalent to saying \[ \frac{1}{x^6} \]. Negative exponents can help simplify complex fractions and make equations easier to work with. They also allow us to express very large or very small numbers more concisely.
Exponent Addition
Adding exponents is useful when you multiply like bases. The rule is, if you have the same base, you add the exponents.
For instance, if you multiply \( x^{-6} \) by \( x^5 \), you add the exponents: \[ x^{-6} \times x^5 = x^{-6 + 5} = x^{-1} \].

Adding exponents simplifies expressions and allows us to combine terms effectively. It's a core part of understanding algebraic manipulation.

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Most popular questions from this chapter

Julia found three Web sites that sell 4-inch-square stickers of her favorite band鈥檚 logo. The three sites sell the stickers for different prices, and charge different amounts for shipping. Site \(1 :\) Stickers are 75 each; shipping is \(\$ 4\) for any size order. Site \(2 :\) Stickers are 60\(\phi\) each; shipping is \(\$ 5.50\) for any size order. Site \(3 :\) Stickers are \(\$ 1.25 ;\) shipping is included. a. For each site, write an equation to represent the charge \(C\) for ordering any number of stickers \(s\) b. Graph your three equations on axes like these. Label each graph with its site number. c. Use your graph to answer this question: If Julia wants to order 16 stickers, which site will charge her the least? d. Use your graph to answer this question: If Julia wants to order 10 stickers, which site will charge her the least?

Determine whether the three points in each set are collinear. \((7,-4),(3,-1),(-2,2.75)\)

Rewrite each expression using a single base and a single exponent. $$(-3)^{81} \cdot(-3)^{141}$$

Which of these sets of numbers share numbers with the powers of 2? Explain how you know. a. positive integer powers of 6 b. positive integer powers of 7 c. positive integer powers of 16

Find the indicated roots without using a calculator. $$ \sqrt{0.0064} $$
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