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Chapter 4: Problem 126

Multiply, and write the answer in simplified form. \(\left(\frac{4}{7}\right)^{4}\)

Short Answer

Expert verified
The answer is \( \frac{256}{2401} \).

Step by step solution

01

Understand the Problem

To multiply and simplify the expression \(\frac{4}{7} \)^4, understand that you are raising the fraction \( \frac{4}{7} \) to the power of 4.
02

Apply the Exponentiation to the Numerator and Denominator

When raising a fraction to a power, both the numerator and the denominator must be raised to that power. This means \(\frac{4}{7}\)^4 = \(\frac{4^4}{7^4}\).
03

Calculate the Powers

Compute \(4^4 \) and \(7^4 \). We have: \ 4^4 = 256 \ and \ 7^4 = 2401.
04

Write the Fraction in Simplified Form

Combine the results from Step 3 into a fraction. Therefore, \(\frac{4^4}{7^4}\) becomes \( \frac{256}{2401} \). This fraction is already in its simplest form.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation where a number, called the base, is raised to the power of an exponent. This exponent indicates how many times the base is multiplied by itself. For instance, in the expression \(\frac{4}{7}^4\), the base is \(\frac{4}{7}\), and the exponent is 4. When you raise \(\frac{4}{7}\) to the 4th power, it's like multiplying \(\frac{4}{7}\) by itself four times. This means: \(\left( \frac{4}{7} \times \frac{4}{7} \times \frac{4}{7} \times \frac{4}{7} \right)\). This process simplifies our calculations and makes otherwise complex steps more straightforward.
Fraction Simplification
Simplifying fractions is an essential skill in math. It involves reducing the numerator and the denominator to their smallest possible whole numbers, while still keeping the same value. For example, \(\frac{256}{2401}\) is already in its simplest form because there are no common factors besides 1 that can divide both 256 and 2401. Simplification in fraction means ensuring both numbers are as small as possible without altering the fraction's value. Always check for the greatest common divisor (GCD) to see if you can simplify the fraction further.
Numerator and Denominator
In every fraction, there are two important parts: the numerator and the denominator. The numerator is the number on top, indicating how many parts we have. The denominator is the number on the bottom, showing how many equal parts the whole is divided into. For instance, in \(\frac{256}{2401}\), 256 is the numerator, and 2401 is the denominator. Both of these parts get raised to the power separately. If we take the previous step, \(\frac{4^4}{7^4}\), it means 4 gets raised to the power of 4, and 7 also gets raised to the power of 4. This fundamental understanding helps in correctly performing operations involving fractions.

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Most popular questions from this chapter

In the following exercises, add. $$\frac{3}{8}+\frac{2}{8}$$

Edgar will have to pay \(\$ 3.75\) in tolls to drive to the city. (a) Explain how he can make change from a \(\$ 10\) bill before he leaves so that he has the exact amount he needs. (b) How is Edgar's situation similar to how you subtract \(10-3 \frac{3}{4} ?\)

In the following exercises, find the least common denominator. $$\frac{1}{3} \text { and } \frac{4}{5}$$

In the following exercises, simplify. $$ \frac{18 r}{27 s} $$

In the following exercises, multiply. $$\frac{2}{9} \cdot\left(-\frac{45}{32}\right)$$
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