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Chapter 5: Problem 140

Simplify. $$ \left[\left(5^{-3}\right)^{2}\right]^{-1} $$

Short Answer

Expert verified
The simplified expression is \( 5^{6} \).

Step by step solution

01

Simplify the Inner Exponent

First, simplify the expression inside the bracket \( (5^{-3})^{2} \). To do this, use the power of a power rule \( (a^m)^n = a^{m \times n} \). So, \( (5^{-3})^{2} = 5^{-3 \times 2} = 5^{-6} \).
02

Apply the Outer Exponent

Now apply the outer exponent \( -1 \) to the simplified expression obtained in Step 1 (\(5^{-6} \)). Again, use the power of a power rule: \( (a^m)^n = a^{m \times n} \). So, \( (5^{-6})^{-1} = 5^{-6 \times -1} = 5^{6} \).
03

Simplify to Get the Final Answer

After simplifying, \( 5^{6} \) is the final answer.

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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
Understanding the power of a power rule is crucial when dealing with exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents: \[ (a^m)^n = a^{m \times n} \]. Hence, if you encounter a problem like \[ (5^{-3})^2 \], apply this rule to get \[ 5^{-3 \times 2} = 5^{-6} \]. This simplification makes it easier to work with the expression and reduces the clutter, helping you see the core values you鈥檙e dealing with.
Negative Exponents
Here鈥檚 an essential rule for working with negative exponents: \[ a^{-n} = \frac{1}{a^n} \]. This means a negative exponent denotes a reciprocal. For instance, \[ 5^{-6} \] can be rewritten as \[ \frac{1}{5^6} \]. Mastering this concept helps you switch between different representations of numbers. It鈥檚 especially useful when you鈥檙e tasked with simplifying complex expressions. Remember, the key is always to turn the negative exponents into positives by placing them in a fraction.
Simplifying Expressions
Simplifying expressions involves breaking them down into more manageable parts and applying rules step by step. Use the power of a power rule and negative exponent rule effectively for this purpose. For example, let's revisit \[ \left(\left(5^{-3}\right)^2\right)^{-1} \]. First, simplify inside the brackets: \[ \left(5^{-3}\right)^2 = 5^{-6} \]. Then, handle the outer exponent: \[ \left(5^{-6}\right)^{-1} = 5^{-6 \times -1} = 5^{6} \]. Each step should follow logically from the previously applied rules, leading to a simplified solution of \[ 5^6 \]. Breaking the expression into these clear steps makes the process more understandable and manageable.

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Most popular questions from this chapter

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