Problem 57 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(y^{8} \cdot y^{-2}\)

Step-by-Step Solution

Answer

The simplified expression is \(y^6\).

Step 1: Apply the Product of Powers Rule

When multiplying powers with the same base, we can add their exponents. So, for the expression \(y^8 \cdot y^{-2}\), we use the rule \(y^m \cdot y^n = y^{m+n}\).

Step 2: Add the Exponents

Add the exponents together: \(8 + (-2) = 6\). This means we simplify the expression to \(y^6\).

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a key concept in algebraic simplification. This rule states that when you multiply two powers that have the same base, you can simply add their exponents together. Imagine the expression \( y^8 \cdot y^{-2} \). Both terms have the same base, which is \( y \). Instead of multiplying the base numbers directly, you apply the rule: for any base \( a \), \( a^m \cdot a^n = a^{m+n} \).
This means you only need to focus on the powers (the exponents) and disregards the base as it stays constant in the calculation. Use this rule whenever you encounter a multiplication of terms with identical bases.

• Identify if the bases are the same.
• Only add the exponents, keeping the base unchanged.
• This dramatically simplifies expressions involving powers of the same base.

Negative Exponents

Negative exponents are a fascinating aspect of algebra. They may seem confusing, but they follow a straightforward logic. A negative exponent indicates that the base should be taken as its reciprocal raised to the corresponding positive power. For example, \( y^{-n} \) is equivalent to \( \frac{1}{y^n} \).
In the given exercise with \( y^8 \cdot y^{-2} \), the \( y^{-2} \) indicates \( \frac{1}{y^2} \). However, instead of converting it to a fraction, by using the product of powers rule, you directly add the exponents, treating it as subtraction due to its negative sign.
This dual approach prevents the need for fractions and streamlines simplification. Always remember:

• Negative exponents mean reciprocal.
• In multiplication, treat negative exponents as subtraction from positive exponents.

Exponent Addition

Exponent addition is an integral part of algebra when dealing with products of powers. It allows for the seamless simplification of expressions that share a common base. Think of this simple equation: adding exponents directly translates to adding the power levels for efficiency.
For instance, in the expression \( y^8 \cdot y^{-2} \), exponent addition results in \( 8 + (-2) = 6 \). This process transforms the expression into \( y^6 \).

• Make sure the bases are the same before adding exponents.
• Negative exponents are treated like subtraction (e.g., \( x^{-n} \) becomes minus \( n \)).
• The outcome is always a simplified expression with a single exponent for powers with the same base.
• This concept illustrates the power of simplification in algebraic operations.