Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{4} \).

Step 1: Solve Negative Exponent

The expression \( \left(\frac{3}{2}\right)^{-2} \) can be simplified by applying the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( \left(\frac{3}{2}\right)^{-2} = \frac{1}{\left(\frac{3}{2}\right)^2} = \frac{1}{\frac{9}{4}} = \frac{4}{9} \).

Step 2: Multiply Fractions

Now that we have simplified to \( \frac{4}{9} \), we multiply it by the other fraction in the expression: \( \frac{4}{9} \cdot \frac{9}{16} \). Use the rule for multiplying fractions: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \). This gives us \( \frac{4 \times 9}{9 \times 16} \).

Step 3: Simplify the Product

The result from the multiplication step is \( \frac{36}{144} \). We simplify by finding the greatest common divisor of 36 and 144, which is 36. Divide the numerator and the denominator by 36 to get \( \frac{1}{4} \).

Key Concepts

Negative Exponents

When encountering negative exponents, remember that they indicate reciprocal relationships. A negative exponent signifies that you should take the reciprocal of the base and then raise it to the positive of that exponent. For instance, if you have \(a^{-n}\), it can be transformed into \(\frac{1}{a^n}\). This rule helps handle otherwise complex-looking expressions.
In the example provided, \( \left(\frac{3}{2}\right)^{-2} \) becomes \( \frac{1}{\left(\frac{3}{2}\right)^2} \). Calculating \( \left(\frac{3}{2}\right)^2 \), we find the expression equals \( \frac{9}{4} \). The reciprocal of \( \frac{9}{4} \) is \( \frac{4}{9} \). Thus, applying the negative exponent rule simplifies the expression significantly. Understanding negative exponents is crucial, enabling you to tackle larger and more complex mathematical scenarios with confidence.

Fraction Multiplication

Multiplying fractions is straightforward when you know the rule: Multiply the numerators together and the denominators together. For two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is \( \frac{a \cdot c}{b \cdot d} \). Always remember to multiply straight across the top and bottom.
For the given problem, after simplifying the negative exponent to \( \frac{4}{9} \), the expression to solve is \( \frac{4}{9} \cdot \frac{9}{16} \). By multiplying the numerators (4 and 9) and then the denominators (9 and 16), we obtain \( \frac{36}{144} \). Multiplication of fractions requires careful handling of each component, ensuring the product is correct before proceeding to simplification.

Simplifying Fractions

Once you've multiplied fractions, simplification is often necessary for a clean, final result. Simplifying a fraction means reducing it to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). This process ensures that the fraction is presented as a reduced ratio.
In our example, the product from the earlier step is \( \frac{36}{144} \). Determine the GCD of 36 and 144, which is 36. Divide both numbers by this GCD to simplify \( \frac{36}{144} \) to \( \frac{1}{4} \). Simplifying fractions not only makes them easier to interpret but also is essential in mathematical communication, ensuring clarity and precision. Always double-check your division to confirm that the fraction cannot be further reduced.