Problem 14 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \frac{3^{-2}}{9} $$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{1}{81} \).

Step 1: Evaluate the Exponent

First, simplify the expression by evaluating the exponent in the numerator. The expression given is \( \frac{3^{-2}}{9} \). \(3^{-2}\) is equivalent to \(\frac{1}{3^2}\), which simplifies to \(\frac{1}{9}\).

Step 2: Substitute and Simplify

Substitute \(\frac{1}{9}\) for \(3^{-2}\) in the original expression. Now, the expression becomes \(\frac{\frac{1}{9}}{9}\), which simplifies to \(\frac{1}{9} \times \frac{1}{9}\).

Step 3: Perform the Multiplication

Multiply the fractions: \(\frac{1}{9} \times \frac{1}{9}\). This results in \(\frac{1 \times 1}{9 \times 9} = \frac{1}{81}\).

Key Concepts

Negative Exponents

Understanding negative exponents is crucial to solving expressions like \( \frac{3^{-2}}{9} \). A negative exponent means you are dividing by that number instead of multiplying. For instance, \( 3^{-2} \) flips the base and turns the exponent positive, resulting in \( \frac{1}{3^2} \).

Why this change, you ask? It's all about creating fractions. A negative in the exponent shifts the number from the numerator to the denominator, or vice-versa. This is a compact way to represent division and can make calculation easier.

• \( x^{-n} = \frac{1}{x^n} \)
• \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)

Always switch the base from the numerator to the denominator, making the exponent positive. In our exercise, turning \( 3^{-2} \) into \( \frac{1}{9} \) simplifies the initial expression greatly.

Fraction Simplification

When working with fractions, simplifying them is a key step to finding an accurate solution. After converting the negative exponent, the expression becomes \( \frac{\frac{1}{9}}{9} \). Simplifying complex fractions like this involves understanding mixed operations.

Think of a complex fraction as a stacked operation. You're essentially performing division between the fractions, which converts into a multiplication process later. The main goal is to make the fraction easier to handle by eliminating unnecessary parts.

• \( \frac{\frac{a}{b}}{c} = \frac{a}{b} \times \frac{1}{c} \)

This transforms to \( \frac{1}{9} \times \frac{1}{9} \). Simplifying by multiplication makes it more straightforward to get to the final answer. Always aim for simplicity.

Multiplication of Fractions

Multiplying fractions is a straightforward but vital arithmetic operation. In our exercise, once we simplified the complex fraction \( \frac{\frac{1}{9}}{9} \) into \( \frac{1}{9} \times \frac{1}{9} \), the process becomes simple multiplication.To multiply fractions:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

Using our example:\[ \frac{1}{9} \times \frac{1}{9} = \frac{1 \times 1}{9 \times 9} = \frac{1}{81} \]Remember to multiply straight across. The result gives you the final answer as the fractions are reduced through multiplication. This operation effectively completes the simplification of the problem while retaining accuracy.