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Chapter 10: Problem 39

Simplify. Write the answer with positive exponents only. $$(2 x)^{-3}$$

Short Answer

Expert verified
\(\frac{1}{8x^3}\)

Step by step solution

01

- Understand Negative Exponents

Recall that a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, \((a^b)^{-1} = \frac{1}{a^b}\).
02

- Apply the Reciprocal Rule

Using the rule from Step 1, rewrite \((2x)^{-3}\) as \(\frac{1}{(2x)^3}\).
03

- Expand the Expression

Now expand the denominator by raising both 2 and x to the power of 3: \(\frac{1}{(2^3 \times x^3)} = \frac{1}{8x^3}\).
04

- Write the Final Answer with Positive Exponents

The simplified expression with positive exponents is \(\frac{1}{8x^3}\).

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Key Concepts

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

Negative Exponents
When you encounter a negative exponent, it means you need to take the reciprocal of the base and change the exponent to a positive value. For instance, \(a^{-b} = \frac{1}{a^b}\). This tells us that instead of multiplying by the base raised to a negative power, we divide by the base raised to a positive power. Simplifying negative exponents makes expressions easier to handle and allows for a clearer understanding of the calculations.
Reciprocal Rule
The reciprocal of a number or expression is simply one divided by that number or expression. When you see a negative exponent like in \( (2x)^{-3} \), you apply the reciprocal rule by rewriting it as \( \frac{1}{(2x)^3} \). This step is crucial because it converts negative exponents into positive ones, making further operations simpler and more straightforward. Remember, always write expressions with positive exponents in your final answer.
Positive Exponents
Positive exponents indicate standard multiplication. For example, \(a^3 = a \times a \times a\). In the context of our problem, after applying the reciprocal rule, we raised \(2x \) to the power of 3: \( (2x)^3 = 2^3 \times x^3 = 8x^3 \). Always remember to expand fully when simplifying, as this helps in writing a clear, final expression.
Prealgebra
Prealgebra lays the foundation for more complex math topics you'll encounter later on. Simplifying expressions, like the one in our problem, is a key skill in prealgebra. Understanding how to manipulate exponents, both positive and negative, and applying rules like the reciprocal rule are fundamental. Mastering these basic concepts will make more advanced algebra and higher math courses much easier. When you practice regularly, these steps will become second nature.

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