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Chapter 4: Problem 125

Multiply, and write the answer in simplified form. \(\left(\frac{6}{5}\right)^{4}\)

Short Answer

Expert verified
\( \frac{1296}{625} \)

Step by step solution

01

Identify the Expression

Recognize that the given expression is \ \ \ \ \( \left( \frac{6}{5} \right)^{4} \).
02

Apply the Power to Both Numerator and Denominator

Raise the numerator (6) and the denominator (5) to the power of 4 separately: \ \ \ \ \( \frac{6^{4}}{5^{4}} \).
03

Calculate the Numerator

Calculate \(6^{4}\): \ 6 \times 6 \times 6 \times 6 = 1296. \ Thus, \(6^{4} = 1296\).
04

Calculate the Denominator

Calculate \(5^{4}\): \ 5 \times 5 \times 5 \times 5 = 625. \ Thus, \(5^{4} = 625\).
05

Combine the Results

Combine the results to form the fraction: \( \frac{1296}{625} \).

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Key Concepts

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

Multiplying Fractions
Multiplying fractions might seem tricky at first, but it's actually very straightforward once you get the hang of it. When multiplying two fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, if you have fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), you multiply them as follows: \[\frac{a \times c}{b \times d} \].
Here's a detailed breakdown:
  • Multiply the numerators: a \( \times \) c
  • Multiply the denominators: b \( \times \) d
  • Combine the results into a new fraction: \( \frac{a \times c}{b \times d} \)
This straightforward method makes multiplying fractions easy.
Raising Fractions to a Power
Raising fractions to a power involves applying the exponent to both the numerator and the denominator. In our example, we need to raise \( \left( \frac{6}{5} \right)^{4} \) to the fourth power. You can break it down into simpler steps:
  • Identify the fraction: \( \frac{6}{5} \)
  • Apply the power of 4 to both the numerator and the denominator: \[ \left( \frac{6}{5} \right)^{4} = \frac{6^{4}}{5^{4}} \]
Next, compute the power of each part separately:
  • Calculate the numerator: \( 6^{4} = 1296 \)
  • Calculate the denominator: \( 5^{4} = 625 \)
Combine the results to get your final fraction: \( \frac{1296}{625} \)
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). For instance, let鈥檚 say we have the fraction \( \frac{10}{20} \). To simplify it, follow these steps:
  • Find the greatest common divisor of 10 and 20, which is 10.
  • Divide the numerator and the denominator by this GCD: \( \frac{10 \div 10}{20 \div 10} = \frac{1}{2} \).
In some cases, like our example \( \frac{1296}{625} \), the fraction is already in its simplest form if there are no common divisors other than 1. Simplified fractions are easier to understand and work with.

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Most popular questions from this chapter

In the following exercises, simplify. $$ \left(-15 \frac{5}{6}\right) \div\left(-3 \frac{1}{6}\right) $$

In the following exercises, perform the indicated operation. $$ 6 \frac{2}{5}+7 \frac{3}{5} $$

In the following exercises, perform the indicated operations and simplify. $$-\frac{22}{25}+\frac{9}{40}$$

In the following exercises, solve the equation. $$ n+\frac{9}{11}=\frac{4}{11} $$

In the following exercises, translate and solve. The difference of \(q\) and one-tenth is \(\frac{1}{2}\) .
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