Problem 23 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \left(\frac{3}{2}\right)^{-2} \cdot \frac{9}{16} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{1}{4} \).

Step 1: Simplify the Negative Exponent

The expression involves a term with a negative exponent: \( \left(\frac{3}{2}\right)^{-2} \). To simplify, recall that \( a^{-b} = \frac{1}{a^b} \). Thus, \( \left(\frac{3}{2}\right)^{-2} = \left(\frac{2}{3}\right)^2 \).

Step 2: Evaluate the Positive Exponent

Now compute \( \left(\frac{2}{3}\right)^2 \). This means we need to square both the numerator and the denominator: \( \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \).

Step 3: Multiply the Fractions

Now multiply the simplified term by the other fraction in the expression: \( \frac{4}{9} \cdot \frac{9}{16} \). Recall that multiplying fractions involves multiplying the numerators and the denominators: \( \frac{4 \times 9}{9 \times 16} = \frac{36}{144} \).

Step 4: Simplify the Result

Simplify \( \frac{36}{144} \), which can be reduced by dividing both the numerator and the denominator by their greatest common divisor, 36: \( \frac{36 \div 36}{144 \div 36} = \frac{1}{4} \).

Key Concepts

Fraction Multiplication

Fraction multiplication is all about multiplying the numerators and the denominators. For example, if we have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), multiplying these fractions results in a new fraction: \( \frac{a \times c}{b \times d} \). It's as simple as that! Each term is multiplied in a straight line, which means no cross-multiplication is involved. In our problem, we took \( \frac{4}{9} \) and \( \frac{9}{16} \), and simply multiplied both the top numbers (numerators) together and both the bottom numbers (denominators) together:

• Top: \( 4 \times 9 = 36 \)
• Bottom: \( 9 \times 16 = 144 \)

This gave us \( \frac{36}{144} \), which then needs to be simplified. Always remember to line up your fractions and multiply directly across.

Fraction Simplification

After multiplying fractions, you might end up needing to simplify the resulting fraction to its simplest form. "Simplifying" means reducing the fraction to the lowest possible terms by finding the greatest common divisor (GCD) of the numerator and the denominator.For our case, we ended with \( \frac{36}{144} \). By observing both numbers, they can both be exactly divided by 36.

• Divide the numerator: \( 36 \div 36 = 1 \)
• Divide the denominator: \( 144 \div 36 = 4 \)

This simplifies \( \frac{36}{144} \) to \( \frac{1}{4} \). Simplifying fractions is essential because it gives you the most accurate representation of the value, making it easier to understand and use in further calculations.

Exponent Rules

Understanding exponent rules is crucial when handling expressions, especially those involving negative exponents. A negative exponent indicates reciprocal action. Recall:

• \( a^{-b} = \frac{1}{a^b} \)

So for \( \left(\frac{3}{2}\right)^{-2} \), we translate this as \( \left(\frac{2}{3}\right)^2 \).Once you have a positive exponent \( (\frac{2}{3})^2 \), the task is to square both the numerator and the denominator separately:

• Numerator: \( 2^2 = 4 \)
• Denominator: \( 3^2 = 9 \)

This gives us the result \( \frac{4}{9} \) from the negative exponent portion of our original expression. Using these exponent rules enables us to transition from negative to positive exponents and properly compute expressions.