Problem 91 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{8}{9}\right)^{2}$$

Step-by-Step Solution

Answer

\( \left(\frac{8}{9}\right)^2 = \frac{64}{81} \).

Step 1: Understand the Problem

The given expression is \( \left(\frac{8}{9}\right)^2 \), which means we need to square the fraction \( \frac{8}{9} \). This involves multiplying the fraction by itself.

Step 2: Multiply the Numerators

The numerator of the fraction \( \frac{8}{9} \) is 8. To find the numerator of the squared fraction, calculate \( 8 \times 8 \). This results in 64.

Step 3: Multiply the Denominators

The denominator of the fraction \( \frac{8}{9} \) is 9. To find the denominator of the squared fraction, calculate \( 9 \times 9 \). This results in 81.

Step 4: Form the New Fraction

Combine the results from Steps 2 and 3 to form the new fraction. The squared fraction is \( \frac{64}{81} \).

Step 5: Reduce the Fraction (If Possible)

Check if the fraction \( \frac{64}{81} \) can be reduced further by finding any common factors of 64 and 81. Since 64 is \( 2^6 \) and 81 is \( 3^4 \), there are no common factors apart from 1. Thus, the fraction is already in its simplest form.

Key Concepts

Squaring Fractions

Squaring a fraction involves raising the fraction to the power of two, which simply means multiplying the fraction by itself.
It's a specific case of exponentiation where the exponent is two. For example, consider the fraction \( \frac{8}{9} \). When we square it, we simply multiply \( \frac{8}{9} \) by itself:

• Mathematically, this is represented as \( \left( \frac{8}{9} \right)^2 = \frac{8}{9} \times \frac{8}{9} \).

This results in a new fraction, which we'll calculate by multiplying both the numerator and the denominator of the original fraction separately by themselves.
This is a straightforward operation that can often be confused with just multiplying the fraction by two, but remember that squaring always involves exponentiation.

Numerator

In any fraction, the numerator is the top number. It represents how many parts out of a whole are being considered.
For the fraction \( \frac{8}{9} \), 8 is the numerator. When squaring fractions, you'll need to square the numerator:

• Multiply the numerator by itself: \( 8 \times 8 = 64 \).

This operation scales the portion of the whole being described by that particular value in the fraction.
So, for \( \left( \frac{8}{9} \right)^2 \), the numerator of the squared fraction is 64. This step is crucial because it determines the part of the whole described by the squared fraction, maintaining the relationships defined by the operation.

Denominator

The denominator is the bottom part of the fraction. It shows into how many total parts the whole is divided.
For the fraction \( \frac{8}{9} \), the number 9 is the denominator. When squaring the fraction, you also square the denominator:

• Multiply the denominator by itself: \( 9 \times 9 = 81 \).

This determines how many total parts the scaled portion is divided into in the result.
In the squared fraction \( \left( \frac{8}{9} \right)^2 \), the denominator is 81, showing the recalibration of the fraction's value in part division terms.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1.
Once you obtain a fraction, like \( \frac{64}{81} \), explore the possibility of simplification by finding common factors:

• Factor the numerator: 64 as \( 2^6 \).
• Factor the denominator: 81 as \( 3^4 \).
• There are no common factors other than 1, which means \( \frac{64}{81} \) is already in its simplest form.

In exercises like these, checking for simplification is a vital final step, ensuring that the smallest possible version of the fraction is presented.
Remembering prime factorization helps in verifying that no further simplification is possible.