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Chapter 5: Problem 66

Simplify each expression. Write answers using positive exponents. $$ \frac{1}{4^{-3}} $$

Short Answer

Expert verified
The simplified form is 64.

Step by step solution

01

Understand the Power of a Fraction

The expression given is \( \frac{1}{4^{-3}} \). This expression involves a negative exponent in the denominator. A negative exponent indicates that the base should be taken as the reciprocal and the exponent made positive.
02

Apply the Negative Exponent Rule

For any non-zero number \( a \), \( a^{-n} \) is equal to \( \frac{1}{a^n} \). Therefore, \( 4^{-3} \) is the same as \( \frac{1}{4^3} \). Substituting this into the expression gives us: \[ \frac{1}{\frac{1}{4^3}} = 4^3 \].
03

Calculate the Positive Exponent

Now, calculate \( 4^3 \). This means multiplying 4 by itself three times: \( 4 \times 4 \times 4 = 64 \).
04

Write the Final Simplified Expression

The simplified form of the expression \( \frac{1}{4^{-3}} \) is \( 64 \), written with positive exponents as required.

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Key Concepts

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

Exponent Rules
Exponents are a way to express repeated multiplication of the same number. For example, \( 4^3 \) means \( 4 \times 4 \times 4 \). Understanding how to work with exponents, especially when they are negative, is essential.**Negative Exponent Rule**: This is a crucial rule to understand. When you have a negative exponent, it means you take the reciprocal of the base and use the positive exponent. For example, \( a^{-n} \) is the same as \( \frac{1}{a^n} \). So, \( 4^{-3} \) turns into \( \frac{1}{4^3} \).Remember that these rules apply as long as the base is non-zero. This makes handling negative exponents more straightforward and avoids division by zero.
Simplifying Expressions
Simplifying expressions is all about making them easier to understand or work with. When dealing with exponents, especially negative ones, simplification involves applying the exponent rules effectively.To simplify an expression like \( \frac{1}{4^{-3}} \), we use the negative exponent rule to take the reciprocal. This turns the expression into \( \frac{1}{\frac{1}{4^3}} \), which is the same as \( 4^3 \). Each step reduces complexity and helps achieve a cleaner expression.Through simplification, expressions become less cluttered, and calculations become easier to manage, allowing you to reach solutions with less error.
Positive Exponents
Working with positive exponents is a fundamental skill in mathematics. A positive exponent, like in \( 4^3 \), means you multiply the base number by itself as many times as indicated by the exponent. Here, \( 4^3 \) means \( 4 \times 4 \times 4 \), resulting in 64.It is often important to express answers using positive exponents to meet standard mathematical conventions. This provides clarity and ensures that the expressions are in their simplest and most interpretable form.Positive exponents are straightforward because they directly tell you how many times to multiply the base, making calculations and understanding more intuitive.

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