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Chapter 4: Problem 31

\(4^{-3}\)

Short Answer

Expert verified
\(4^{-3} = \frac{1}{64}\)

Step by step solution

01

Understand the Negative Exponent

Recognize that a negative exponent indicates that the base number should be taken as the reciprocal and the sign of the exponent should be changed. For example, 4 raised to the power of -3 can be written as the reciprocal of 4 raised to the power of 3.
02

Apply the Negative Exponent Rule

Rewrite the expression using the reciprocal: \[ 4^{-3} = \frac{1}{4^3} \]
03

Calculate the Positive Exponent

Calculate \(4^3\), which is 4 multiplied by itself three times. So, \[ 4^3 = 4 \times 4 \times 4 = 64 \]
04

Substitute the Value

Substitute the calculated value back into the equation: \[ 4^{-3} = \frac{1}{64} \]

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Key Concepts

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

Understanding Reciprocals
A reciprocal is defined as the inverse of a number. For any nonzero number, its reciprocal is found by flipping the numerator and the denominator. For instance, the reciprocal of 4 is \(\frac{1}{4}\). When a number has a negative exponent, it means we should use the reciprocal of that number.
To use reciprocals with exponents, we switch the base to its reciprocal and then make the exponent positive.
Introduction to Exponentiation
Exponentiation is a mathematical operation involving a base and an exponent. The exponent tells us how many times to multiply the base by itself. For example, with \(\text{4}^{3}\), we multiply 4 by itself three times. So, \(\text{4}^{3}\=4 \times 4 \times 4 = 64\).
Understanding how to handle negative exponents takes this a step further. As seen in the expression \(4^{-3}\), we calculate the reciprocal first, then raise the base to the power of the positive exponent resulting in \(\frac{1}{4^{3}} = \frac{1}{64}\).
Step-by-Step Solution
For the expression \(4^{-3}\), we can break it down into easy steps:
Step 1: Understand the Negative Exponent - A negative exponent means you take the reciprocal of the base and change the exponent to positive. So, \(4^{-3}\) becomes \(\frac{1}{4^3}\).
Step 2: Apply the Negative Exponent Rule - We rewrite the expression using the reciprocal, which gives us \(\frac{1}{4^3}\).
Step 3: Calculate the Positive Exponent - Next, we calculate the positive exponent: \(4^3 = 4 \times 4 \times 4 = 64\).
Step 4: Substitute the Value - Finally, substitute the calculated value back into the equation: \(4^{-3} = \frac{1}{64}\).
This process helps us understand that the negative exponent dictates taking the reciprocal, transforming the expression to a more familiar positive exponent form.

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Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m+6)^{2} $$

In \(2016, \$ 1.14 \times 10^{10}\) was spent to attend motion pictures in the United States and Canada. The total number of tickets sold was 1.32 billion. What was the average ticket price (to the nearest cent) for a movie? (Data from Motion Picture Association of America.)

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 301 \times 299 $$

Perform each indicated operation. \(\left(8 m^{2}-7 m\right)-\left(3 m^{2}+7 m-6\right)\)

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-4\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$
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