Question

Simplify each expression. \(\left(4^{-3}\right)^{-1}\)

Short answer

64

Step-by-step solution

Apply the Power of a Power Rule

Use the rule \( (a^m)^n = a^{mn} \) to simplify the expression. In this case, \(4^{-3}\) is raised to the power of \(-1\). Multiply the exponents: \( (4^{-3})^{-1} = 4^{(-3) \times (-1)} \).

Simplify the Exponents

Perform the multiplication of the exponents. \((-3) \times (-1)\) equals \3\. Therefore, \(4^{(-3) \times (-1)} = 4^3 \).

Evaluate the Simplified Expression

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

Key concepts

Power of a Power Rule

When dealing with exponents, it's crucial to understand the \textbf{Power of a Power Rule}. This rule is used when an exponent is raised to another exponent, like \textbf{\( (a^m)^n \)}. The rule states that you multiply the exponents together:
\[ (a^m)^n = a^{mn} \]
In our exercise, we started with \textbf{\( (4^{-3})^{-1} \)}. By applying the Power of a Power Rule, we multiply the exponents \textbf{-3} and \textbf{-1} together:
\[ (4^{-3})^{-1} = 4^{(-3) \times (-1)} \]
This simplifies to \textbf{\( 4^3 \)}. Using this rule makes it much easier to work with exponents and simplifies complex expressions quickly.

Simplifying Exponents

Simplifying exponents involves performing arithmetic operations on the exponents to reduce the expression to a simpler form. In our case, after applying the Power of a Power Rule, we reached \textbf{\( 4^{(-3) \times (-1)} \)}. Here’s the process to simplify it:
\[ (-3) \times (-1) = 3 \]
When we multiply the exponents \textbf{-3} and \textbf{-1}, we get \textbf{3}. Therefore, our expression simplifies to:
\[ 4^3 \]
Understanding how to simplify exponents by performing basic arithmetic like multiplication is an essential skill. It’s especially useful in algebra and higher-level math problems where exponent rules are frequently applied.

Evaluating Exponential Expressions

Evaluating exponential expressions means calculating the final value of the given expression. After simplifying our expression to \textbf{\( 4^3 \)}, the next step is to calculate its value. To do this, you multiply the base \textbf{4} by itself three times:
\[ 4 \times 4 \times 4 \]
\( 4 \times 4 = 16 \)
Then, we multiply that result by \textbf{4} again:
\( 16 \times 4 = 64 \)
So, \textbf{\( 4^3 = 64 \)}. Evaluating exponential expressions can often be done with a calculator but understanding how to do it manually reinforces the basics of exponents and roots, making you more confident in tackling more complex problems in mathematics.