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Chapter 1: Problem 23

Find the value of each exponential expression. $$ \left(\frac{2}{3}\right)^{4} $$

Short Answer

Expert verified
The value is \( \frac{16}{81} \).

Step by step solution

01

Understand the problem

The task is to find the value of the exponential expression \(\bigg(\frac{2}{3}\bigg)^{4}\). This means raising the fraction \(\frac{2}{3}\) to the power of 4.
02

Apply the exponent to both numerator and denominator

Since the exponent applies to both the numerator and the denominator, rewrite the expression as \(\frac{2^{4}}{3^{4}}\).
03

Calculate the powers

Compute the powers of the numerator and the denominator separately. \(2^{4} = 2 \times 2 \times 2 \times 2 = 16\) and \(3^{4} = 3 \times 3 \times 3 \times 3 = 81\).
04

Combine the results

Combine the results of the previous calculations to find the value of the expression. \(\frac{2^{4}}{3^{4}} = \frac{16}{81}\).

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Key Concepts

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

headline of the respective core concept
Exponentiation is a mathematical operation, often represented using the symbol '^'. It involves raising a base number to a power or exponent. For example, in the expression \(2^{4}\), 2 is the base, and 4 is the exponent. The exponent tells you how many times to multiply the base by itself.
In this particular exercise, we have \(\bigg(\frac{2}{3}\bigg)^{4}\). The base here is the fraction \(\frac{2}{3}\), and the exponent is 4. The exponentiation means we must multiply \(\frac{2}{3}\) by itself four times. This operation can be performed more easily by dealing with the numerator and denominator separately, which brings us to our next concept.
headline of the respective core concept
In a fraction, the top number is called the 'numerator,' and the bottom number is the 'denominator.' The fraction \(\frac{2}{3}\) has 2 as its numerator and 3 as its denominator. When dealing with exponentiation involving fractions, each part of the fraction is handled separately under the exponent.
For instance, in our exercise, we can rewrite \(\bigg(\frac{2}{3}\bigg)^{4}\) as \(\frac{2^{4}}{3^{4}}\) by applying the exponent to both the numerator and the denominator. This step simplifies the process significantly, making the calculation more straightforward. Understanding this separation will help you solve fractional exponentiation problems more easily.
headline of the respective core concept
Powers (or exponents) indicate how many times to multiply the base by itself. Calculating powers is essential in many areas of mathematics.
In the expression \(2^{4}\), you multiply 2 by itself four times: \2 \times 2 \times 2 \times 2 = 16\. Similarly, for \(3^{4}\), you multiply 3 by itself four times: \3 \times 3 \times 3 \times 3 = 81\.
So, \(\frac{2^{4}}{3^{4}}\) simplifies to \(\frac{16}{81}\). Breaking down the calculation into these steps helps you understand each part, ensuring you get the correct answer.

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Most popular questions from this chapter

In September, Jeffery Cooper began with a checking account balance of 904.89 Dollar. His checks and deposits for September are as follows: $$ \begin{array}{|l|c|} \hline \text { Checks } & \text { Deposits } \\ \$ 41.29 & \$ 80.59 \\ \$ 13.66 & \$ 276.13 \\ \$ 84.40 & \\ \hline \end{array} $$ Assuming no other transactions, what was his account balance at the end of September?

Perform each indicated operation. $$ \left(-\frac{1}{2}+0.25\right)-\left(-\frac{3}{4}+0.75\right) $$

Perform each indicated operation. $$ |-4+8|+|6-1| $$

Solve each problem. The largest change in temperature ever recorded within a 24 -hr period occurred in Montana, on January \(23-24,1916\). The temperature fell \(100^{\circ} \mathrm{F}\) from a starting temperature of \(44^{\circ} \mathrm{F} .\) What was the low temperature during this period?

To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. For example, to find the average of \(14,8,3,9,\) and \(1,\) we add them and then divide by 5. $$ \frac{14+8+3+9+1}{5}=\frac{35}{5}=7 \leftarrow \text { Average } $$ Find the average of each group of numbers. \(-15,29,8,-6\)
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