Problem 84 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{60 y^{6}}{80 y^{2}} $$

Step-by-Step Solution

Answer

\( \frac{3}{4}y^4 \)

Step 1: Simplify the Fraction

First, simplify the numerical fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. For 60 and 80, the GCD is 20. Divide both numbers by their GCD. \[ \frac{60}{80} = \frac{60 \div 20}{80 \div 20} = \frac{3}{4} \]

Step 2: Simplify the Exponents

Next, simplify the variable part of the expression by subtracting the exponents. According to the laws of exponents, when dividing two exponents with the same base, we subtract the exponents.Given \( y^6 \div y^2 \), subtract the exponent of the denominator from the exponent of the numerator:\[ y^{6-2} = y^4 \]

Step 3: Combine Results

Combine the simplified fraction from Step 1 with the simplified exponent result from Step 2. The expression now becomes:\[ \frac{3}{4}y^4 \]

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a fundamental concept when simplifying expressions, particularly fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Recognizing the GCD helps reduce fractions to their simplest form.

• For example, consider the numbers 60 and 80. The GCD of these two numbers is 20.
• By dividing both by 20, you simplify the fraction from \( \frac{60}{80} \) to \( \frac{3}{4} \).

Using the GCD is a powerful tool in simplifying fractions, making it easier to work with them in mathematical expressions. This process not only simplifies expressions
but also facilitates further calculations that may be needed down the road.

Laws of Exponents

Understanding the laws of exponents is essential when dealing with expressions involving powers. These laws provide rules for performing operations on expressions with exponents, ensuring that the calculations are accurate.

• When dividing exponents with the same base, subtract the exponents: \( a^m / a^n = a^{m-n} \).
• For example, \( y^6 / y^2 \) leads to \( y^{6-2} = y^4 \).

Applying these laws helps you manage expressions more effectively. They simplify calculations significantly, allowing you to reduce complex expressions to simpler, more manageable forms.
Always remember the basic rules:

• Product of Powers Rule: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power Rule: \( (a^m)^n = a^{mn} \).

These laws are fundamental in algebra and critical for simplifying expressions.

Fraction Simplification

Fraction simplification is a crucial step in solving mathematical problems, especially in algebra. It involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their GCD.

• Identify the GCD of the numerator and denominator.
• Use this GCD to divide both parts of the fraction.

For example, in simplifying \( \frac{60}{80} \), the GCD is 20, simplifying the fraction to \( \frac{3}{4} \).
Simplification makes fractions easier to work with by eliminating common factors.
It is also a helpful tool in breaking down complex problems into simpler parts. Through simplification, you can uncover relationships between different parts of an equation or expression.