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Chapter 4: Problem 124

Multiply, and write the answer in simplified form. \(\left(\frac{4}{5}\right)^{2}\)

Short Answer

Expert verified
The simplified form is \(\frac{16}{25}\).

Step by step solution

01

- Understand the Problem

The problem requires finding the square of the fraction \(\frac{4}{5}\).
02

- Apply the Exponent

To square the fraction, we need to raise both the numerator and the denominator to the power of 2. Write it as: \(\frac{4^2}{5^2}\).
03

- Compute the Numerator

Calculate the square of the numerator: \(4^2 = 16\).
04

- Compute the Denominator

Calculate the square of the denominator: \(5^2 = 25\).
05

- Write the Simplified Form

Combine the squared numerator and denominator to write the fraction: \(\frac{16}{25}\).

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Key Concepts

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

Squaring Fractions
Squaring a fraction involves multiplying the fraction by itself. When you square a fraction, both the numerator (the top number) and the denominator (the bottom number) must be squared. For example:
Let's take the fraction \(\frac{4}{5}\). To square this fraction, we need to apply the exponent of 2 to both the numerator and the denominator.

You write it as: \(\frac{4^2}{5^2}\)

This will give us \(\frac{16}{25}\), since \({4^2 = 16}\) and \({5^2 = 25}\).

Remember, squaring a fraction does not change the operation you need to perform. You simply apply the exponent to each part of the fraction individually!
Simplifying Fractions
After performing operations like squaring a fraction, it鈥檚 always a good idea to check if you can simplify the result. Simplifying a fraction means reducing it to its lowest possible form.
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCD.
For example, the fraction \(\frac{16}{25}\) from our earlier steps is already in its simplest form, because 16 and 25 have no common factors other than 1.

However, if you had a fraction like \(\frac{8}{20}\), you could simplify it by finding the GCD, which is 4 in this case, and then divide both the numerator and the denominator by 4:

\(\frac{8 \div 4}{20 \div 4} = \frac{2}{5}\)

Simplifying fractions makes them easier to work with and helps in comparing them more easily.
Numerator and Denominator
Understanding numerators and denominators is key to mastering fractions.
  • Numerator: This is the top number of a fraction. It represents how many parts of a whole are being considered.
  • Denominator: This is the bottom number of a fraction. It shows the total number of equal parts the whole is divided into.
Using our example fraction \(\frac{4}{5}\):

  • The number 4 is the numerator. It tells us we are considering 4 parts out of the whole.
  • The number 5 is the denominator. It tells us that the whole is divided into 5 equal parts.
When squaring this fraction, each part (the numerator and the denominator) is squared separately to get \(\frac{4^2}{5^2}\), resulting in \(\frac{16}{25}\).

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Most popular questions from this chapter

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x+\frac{2}{3}\) when (a)\(x=-\frac{1}{6} \quad\) (b) \(x=-\frac{5}{3}\)

In the following exercises, perform the indicated operation. $$ 3 \frac{5}{8}+3 \frac{7}{8} $$

Translate and solve. The sum of two-thirds and \(n\) is \(-\frac{3}{5}\)

In the following exercises, simplify. $$ \frac{\frac{5}{14}+\frac{1}{8}}{\frac{9}{56}} $$

In the following exercises, subtract. $$\frac{8}{11}-\frac{6}{11}$$
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