Problem 14 - Elementary Algebra

Question

Find each quotient $$ \frac{y^{9}}{y^{5}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\frac{y^9}{y^5}\) Answer: \(y^4\)

Step 1: Identify the property of exponents to apply

We will use the quotient rule of exponents, which states that for any nonzero number \(a\) and integers \(m\) and \(n\): $$ a^m \div a^n = a^{(m-n)} $$ In our case, the base is \(y\) and the exponents are \(9\) and \(5\).

Step 2: Apply the quotient rule of exponents

Using the quotient rule for exponents, we have: $$ \frac{y^9}{y^5} = y^{(9-5)} $$

Step 3: Simplify the expression

Subtracting the exponents, we get: $$ y^{(9-5)} = y^4 $$ The simplified quotient is \(y^4\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a quantity or a relationship. An example of an algebraic expression is \( y^9 \), where \( y \) is a variable and \( 9 \) is the exponent.Variables, like \( x \) or \( y \), stand for numbers whose values can change or are yet to be determined. Constants are fixed numerical values. Operators like '+' or '\( \times \)' and exponents are used to link the numbers and variables within an expression.
When working with algebraic expressions containing exponents, it's essential to understand how to manipulate them using different rules. This manipulation allows us to rearrange and simplify expressions to make them easier to work with. The quotient rule for exponents is one of the key rules that aids in this simplification process.

Exploring Exponents

Exponents are a fundamental part of algebra. They indicate how many times a number, known as the base, is multiplied by itself. In other words, in an expression like \( y^9 \), \( y \) is the base, and \( 9 \) is the exponent, which means \( y \) is multiplied by itself 9 times.

• Positive Exponents: Repeat the base multiplication. E.g., \( y^3 = y \times y \times y \).
• Negative Exponents: Represent division by the base. E.g., \( y^{-2} = \frac{1}{y^2} \).
• Zero Exponent Rule: Any nonzero base raised to the power of zero equals one. E.g., \( y^0 = 1 \).

These rules make it easier to handle expressions with exponents, particularly when simplifying them or performing arithmetic operations. Knowing how exponents function is crucial for the correct application of the quotient rule.

Simplifying Expressions with the Quotient Rule of Exponents

Simplifying expressions is a key skill in algebra that involves reducing expressions to their simplest form. This often means combining like terms and applying rules of exponents. The quotient rule of exponents defines how to divide expressions that have the same base.
According to this rule, when you divide two expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). For example, in the expression \( \frac{y^9}{y^5} \), both terms have the base \( y \).

• Identify the base: Here, it's \( y \).
• Subtract the exponents: \( 9 - 5 = 4 \).
• Therefore, \( \frac{y^9}{y^5} = y^4 \).

This process highlights how important it is to understand and apply the basic rules of exponents, enabling you to simplify even complex algebraic expressions effectively.