Problem 35 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ y^{5} y^{7} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$y^{5} y^{7}$$. Answer: $$y^{12}$$

Step 1: Identify the base and exponents

The expression is: $$ y^{5} y^{7} $$ Here, the base is "y" and the exponents are 5 and 7.

Step 2: Apply the Product Rule

Using the Product Rule, we can add the exponents with the same base: $$ y^{5} y^{7} = y^{5+7} $$

Step 3: Simplify the Expression

Now, let's add the exponents: $$ y^{5+7} = y^{12} $$ The simplified expression is: $$ y^{12} $$

Key Concepts

Product Rule of Exponents

When simplifying expressions with exponents, a fundamental rule applied is the product rule. This rule states that when you multiply two expressions with the same base, you can add the exponents. For example, for the base 'y' and exponents 5 and 7, as in the expression \( y^{5} \times y^{7} \), we apply the product rule.

• Identify the base and the exponents: Here it is 'y' and 5, 7 respectively.
• Apply the rule: \( y^{5} \times y^{7} = y^{5+7} \)
• Simplify the sum of the exponents: \( y^{12} \).

This rule is vital for combining expressions during multiplication and is a cornerstone of algebraic simplification.

Quotient Rule of Exponents

Similar to the product rule, the quotient rule of exponents is another essential principle in algebra when you divide expressions that have the same base. According to the quotient rule, for a base 'y' and exponents 'm' and 'n', the expression \( \frac{y^m}{y^n} \) simplifies to \( y^{m-n} \).
This rule demonstrates that you can subtract the exponent in the denominator from the exponent in the numerator. By doing this, you simplify the expression significantly, allowing for clearer and more concise answers in algebraic problems.
For instance, if you have \( \frac{y^{7}}{y^{2}} \), it simplifies to \( y^{7-2} = y^{5} \), reducing the original complex expression to a more manageable form.

Exponent Properties

Beyond the product and quotient rules, additional exponent properties are crucial for simplifying algebraic expressions. These properties include:

• Power of a power: When raising an exponent to another exponent, multiply the exponents. \( (y^m)^n = y^{mn} \).
• Power of a product: To raise a product to a power, raise each factor to the power separately. \( (ab)^n = a^n b^n \).
• Zero exponent: Any base with an exponent of zero equals one. \( y^0 = 1 \).
• Negative exponent: A negative exponent indicates a reciprocal. \( y^{-n} = \frac{1}{y^n} \).

Understanding and applying these properties allows students to confidently tackle more complex problems involving exponents.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form using a combination of arithmetic and algebraic principles, including the rules of exponents. The process often includes:

• Combining like terms
• Applying the distributive property
• Factoring and expanding expressions
• Using exponent rules to simplify exponential terms

For example, in the expression \( y^{5} y^{7} \), simplification is achieved by adding exponents, since the terms share the same base. But algebraic simplification can become more complex as variables and polynomial terms are included, making these foundational rules essential for a student’s toolkit in mathematical problem-solving.
The primary goal of algebraic simplification is to make expressions easier to understand and work with, which is especially important when solving for variables in equations.