Problem 87 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$\left(\frac{5}{6}\right)^{2}$$

Step-by-Step Solution

Answer

\( \left(\frac{5}{6}\right)^{2} = \frac{25}{36} \)

Step 1: Understand the Problem

We need to evaluate the expression \( \left(\frac{5}{6}\right)^{2} \), which means we have to square the fraction \( \frac{5}{6} \).

Step 2: Square the Numerator

The numerator of the fraction is 5. To square it, calculate \( 5^2 \). This results in 25.

Step 3: Square the Denominator

The denominator of the fraction is 6. To square it, calculate \( 6^2 \). This gives us 36.

Step 4: Form the New Fraction

After squaring the numerator and the denominator separately, we form the new fraction: \( \frac{25}{36} \).

Step 5: Simplify the Fraction

Check if the fraction \( \frac{25}{36} \) can be simplified by finding common factors between 25 and 36. Since they have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Numerator

In any given fraction, the numerator is the top part. It indicates how many parts of the whole are being considered or counted. For instance, in the fraction \( \frac{5}{6} \), the numerator is 5. This tells us that we are considering or taking 5 parts of a divided whole.

• The numerator sits above the line in a fraction.
• It can be any integer, positive or negative.
• When dealing with operations like addition, subtraction, multiplication, or division, understanding what the numerator represents helps simplify and solve the problem correctly.

Simplifying fractions sometimes involves adjusting both the numerator and the denominator to find a common factor, but the core concept remains: the numerator shows how many parts we're interested in.
Whenever you square a fraction, you need to square the numerator as part of the process.

Denominator

The denominator in a fraction serves as the base or "whole" part, indicating the number of equal parts the whole is divided into. In the fraction \( \frac{5}{6} \), the denominator is 6, indicating that the whole is divided into 6 equal parts.

• The denominator provides context to the numerator's count of parts.
• It rests below the line in a fraction.
• The denominator can also be any positive or negative integer, but never zero, since division by zero is undefined.

When squaring a fraction, just like the numerator, the denominator must also be squared. Understanding the denominator's role helps in grasping the fraction's size and how it relates to a whole unit.
For the fraction \( \frac{5}{6} \), you square the denominator by calculating \( 6^2 \), resulting in 36.

Fraction Simplification

Fraction simplification is the process of reducing a fraction by removing common factors until it reaches its simplest form. A fraction is considered simple when no smaller, whole numbers can evenly divide both the numerator and the denominator (other than 1).

• Identify the greatest common divisor (GCD) for the numerator and denominator.
• Divide both the numerator and denominator by the GCD to simplify.
• A simplified fraction is easier to interpret and often necessary for further mathematical operations.

Even though in the exercise \( \frac{25}{36} \) appears at first glance complex, upon checking, both numbers have no common divisor other than 1. Therefore, the fraction is in its simplest form, which simplifies reading and understanding of the represented quantity.

Exponents

Exponents are a mathematical shorthand used to represent repeated multiplication. When you see a number raised to an exponent, it tells you to multiply that number by itself as many times as indicated by the exponent. In \( \left(\frac{5}{6}\right)^{2} \), the 2 is the exponent, telling us to multiply \( \frac{5}{6} \) by itself.

• Exponents indicate repeated multiplication of the same number or value.
• In fractions, exponents apply separately to both the numerator and the denominator when squaring.
• The expression \( a^n \) means \( a \times a \times \cdots \times a \) (n times).

When squaring a fraction, both the numerator and denominator are raised to the power of 2. This process keeps the proportion of the fraction intact while expanding its numerical value. Understanding exponents is crucial in recognizing how powers affect the size and value of numbers and fractions in various calculations.