Problem 11 - Elementary and Intermediate Algebra

Question

Complete each solution to simplify each expression. \(\left(y^{5} y^{3}\right)^{-5}=(\quad)^{-5}=y \quad=\frac{1}{y}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{y^{40}}\).

Step 1: Simplify the Expression Inside the Parentheses

The expression inside the parentheses is \(y^5 y^3\). To simplify this, apply the product of powers rule, which states that when you multiply powers with the same base, you add the exponents: \(y^5 \cdot y^3 = y^{5+3} = y^8\).

Step 2: Substitute the Simplified Expression

Now substitute \(y^8\) back into the expression: \((y^8)^{-5}\).

Step 3: Apply the Power Rule for Exponents

Use the power of a power rule, which states that when raising a power to another power, multiply the exponents: \((y^8)^{-5} = y^{8 \cdot (-5)} = y^{-40}\).

Step 4: Convert to a Reciprocal

An expression with a negative exponent \(y^{-40}\) can be converted to its reciprocal with a positive exponent: \(y^{-40} = \frac{1}{y^{40}}\).

Key Concepts

Product of Powers Rule

When working with algebraic expressions, one essential rule to remember is the **Product of Powers Rule**. This rule makes it straightforward to simplify expressions when you're multiplying powers with the same base.

Here's how it works: if you have two expressions with the same base, like +y^a+ and +y^b+, you can simplify them by adding their exponents. This means that \( y^a \times y^b = y^{a+b} \).

For instance, simplify \( y^5 \times y^3 \):

• First, ensure the bases are the same, which they are in this case (+y+).
• Next, simply add the exponents: \( 5+3 \), giving you \( y^8 \).

This simple addition of exponents is a handy tool, especially when dealing with complex expressions, as it reduces your work and helps prevent mistakes by decreasing the number of steps you need.

Power of a Power Rule

The next rule we'll explore is the **Power of a Power Rule**. This rule simplifies expressions where a power is raised to another power, such as \( (y^a)^b \).

To apply this rule, multiply the exponents: \( (y^a)^b = y^{a \cdot b} \). It ensures that even when expressions seem more complex, you can manage them efficiently.

Consider the expression \( (y^8)^{-5} \). Here's how you simplify it using this rule:

• Identify the base, which is +y+, and the exponents, which are +8+ and +(-5)+.
• Multiply the exponents together: \( 8 \times (-5) = -40 \).
• The simplified expression becomes \( y^{-40} \).

Applying the Power of a Power Rule saves time and transforms challenging, nested powers into simpler ones. With practice, you'll find it contributes significantly to your algebra toolkit.

Negative Exponents

Negative exponents can be puzzling at first, but once understood, they become quite logical. A **Negative Exponent** indicates that we take the reciprocal of the base raised to the opposite positive exponent.

In simpler terms, \( y^{-a} = \frac{1}{y^a} \). This transformation from a negative to a positive exponent involves flipping the base into a fraction.

Let's break down an example using \( y^{-40} \):

• Recognize the negative exponent, which is -40.
• Convert it to a positive by taking the reciprocal: \( y^{-40} \) becomes \( \frac{1}{y^{40}} \).

Understanding this rule is crucial because it allows you to rewrite expressions with negative exponents in a form that's often easier to interpret and work with in further computations.