Problem 66 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\left(y^{-3}\right)^{2}\)

Step-by-Step Solution

Answer

\(\frac{1}{y^6}\)

Step 1: Apply the Power of a Power Rule

When you have an expression raised to a power inside another exponent, like \( (a^m)^n \), you use the power of a power rule: multiply the exponents together. For \( (y^{-3})^2 \), calculate \((-3) \times 2\).

Step 2: Multiply the Exponents

Calculate the multiplication: \((-3) \times 2 = -6\). So, \( (y^{-3})^2 = y^{-6} \).

Step 3: Remove the Negative Exponent by Using the Reciprocal

To eliminate a negative exponent, rewrite the expression with a positive exponent by taking the reciprocal. \(y^{-6}\) becomes \(\frac{1}{y^6}\).

Key Concepts

Negative Exponents

Negative exponents may appear puzzling at first, but understanding them can simplify many mathematical expressions. A negative exponent indicates the reciprocal, or the inverse, of a number raised to a positive exponent.
When you see something like \(a^{-n}\), it's equivalent to \(\frac{1}{a^n}\). This means you "flip" the base into the denominator and change the exponent to a positive.

• Example: \(y^{-3} = \frac{1}{y^3}\)
• It turns division into multiplication and vice versa.

Recognizing when to switch between a negative and positive exponent is an essential skill that makes simplifying expressions much more straightforward.

Power of a Power Rule

The power of a power rule is a handy tool when dealing with exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. Typically, you will see this in expressions like \((a^m)^n = a^{m \times n}\).
For instance, let's consider \((y^{-3})^2\). Applying the rule:

• Step 1: Multiply the exponents: \((-3) \times 2 = -6\)
• Step 2: The result becomes \(y^{-6}\)

Using the power of a power rule is critical for efficiently breaking down and simplifying complex exponentials.

Simplifying Expressions

Simplifying expressions involves making them easier to read or solve, often by reducing them to a simple form or replacing them with an equivalent expression. By applying rules such as the power of a power or understanding negative exponents, you can simplify almost any expression.
Follow these steps:

• Recognize patterns and apply the corresponding rules.
• In \((y^{-3})^2\), apply the power of a power rule first to simplify \(y^{-3}\) to \(y^{-6}\).
• Finally, eliminate negative exponents: convert \(y^{-6}\) into \(\frac{1}{y^6}\).

Mastering simplification techniques makes solving mathematical problems much faster and less fraught with errors.