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Chapter 5: Problem 97

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ -4 r^{-2}\left(r^{4}\right)^{2} $$

Short Answer

Expert verified
-4 r^6

Step by step solution

01

Simplify the power of a power

First, simplify the expression \((r^{4})^{2}\). Use the power of a power rule, which states that \((a^m)^n = a^{mn}\). Thus, \((r^{4})^{2} = r^{4 \cdot 2} = r^8\).
02

Multiply the expressions

Now, multiply the simplified term with the remaining part of the expression. So we have \(-4 r^{-2} \cdot r^8\).
03

Combine the exponents

Use the rule of multiplying like bases, which states that \(a^m \cdot a^n = a^{m+n}\). Therefore, combine the exponents of \(r\) to obtain \(-4 r^{-2 + 8} = -4 r^6\).
04

Final Answer

The expression simplifies to \(-4 r^6\).

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Key Concepts

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

Power of a Power Rule
When you see an expression where an exponent is raised to another exponent, like \((a^m)^n\), you need to use the power of a power rule. This rule lets you combine the two exponents by multiplying them together. For example, \((r^4)^2\) turns into \[r^{4 \cdot 2} = r^8\].
This rule helps simplify the expression so you can more easily work with it later.
Remember: If you have \(a^{m}\), and it's inside another exponent \((n)\), the formula will be \((a^m)^n = a^{m \cdot n}\).
This step is crucial in breaking down complex expressions and making them simpler.
Multiplying Like Bases
Now that you understand the power of a power rule, let's move on to multiplying like bases. When you multiply terms with the same base, you add their exponents together. For example, in the term \(-4 r^{-2} \cdot r^8\), the bases are the same (both are \

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Most popular questions from this chapter

Two expressions are given. Replace x with 3 and y with 4 to show that, in general, the two expressions are not equivalent. $$ (x+y)^{3} ; \quad x^{3}+y^{3} $$

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