Problem 24 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \left(\frac{1}{2}\right)^{4}\left(\frac{5}{2}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{100} \).

Step 1: Simplify the expression using power rules

Start by simplifying the expression \( \left(\frac{1}{2}\right)^{4}\left(\frac{5}{2}\right)^{-2} \) using the power rule \( a^{-n} = \frac{1}{a^n} \). Rewrite the expression as \( \left(\frac{1}{2}\right)^{4} \times \left(\frac{2}{5}\right)^{2} \).

Step 2: Calculate each power separately

Calculate \( \left(\frac{1}{2}\right)^{4} \) by multiplying \( \frac{1}{2} \) by itself four times: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16} \). Calculate \( \left(\frac{2}{5}\right)^{2} \) by multiplying the fraction by itself: \( \frac{2}{5} \times \frac{2}{5} = \frac{4}{25} \).

Step 3: Multiply the results

Multiply the results from Step 2 together: \( \frac{1}{16} \times \frac{4}{25} \). Use the multiplication of fractions method: \( \frac{1 \times 4}{16 \times 25} = \frac{4}{400} \).

Step 4: Simplify the fraction

Simplify \( \frac{4}{400} \) by dividing both the numerator and denominator by their greatest common divisor, which is 4: \( \frac{4 \div 4}{400 \div 4} = \frac{1}{100} \).

Key Concepts

Power Rules

In mathematics, power rules are essential when dealing with exponential expressions. Understanding and using these rules can simplify expressions considerably. For instance, if you see an expression like

• \( a^{-n} \)

the power rule tells us that this can be rewritten as

• \( \frac{1}{a^n} \).

This is crucial when transforming negative exponents into positive ones, allowing us to work in more manageable terms. In the original exercise involving

• \( \left(\frac{5}{2}\right)^{-2} \),

we applied the power rule to rewrite it as

• \( \left(\frac{2}{5}\right)^{2} \).

Mastering such transformations can be quite beneficial, especially when working with complex formulas or multiple variables. Ensuring that you thoroughly understand the power rules will empower you to navigate similar problems with ease.

Fraction Multiplication

Multiplying fractions is a straightforward process that requires multiplying the numerators and the denominators separately. This step is often where students can use their understanding of power rules effectively. In our exercise, after applying the power rules, we found ourselves needing to calculate:

• \( \left(\frac{1}{2}\right)^{4} \)
• \( \left(\frac{2}{5}\right)^{2} \).

Recognizing that each fraction is raised to a power, we multiply:

• \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16} \)
• \( \frac{2}{5} \times \frac{2}{5} = \frac{4}{25} \).

Once the individual powers are computed, as is our next step:

• \( \frac{1}{16} \times \frac{4}{25} \).

We multiply across the numerators and denominators to get the result:

• \( \frac{1 \times 4}{16 \times 25} = \frac{4}{400} \).

Thus, appreciating fraction multiplication mechanics is key, since it appears frequently in problems involving fractions and exponentiation.

Fraction Simplification

Simplification is the final and crucial step when manipulating fractional expressions. The goal is to express a fraction in its simplest or most reduced form by dividing both the numerator and the denominator by their greatest common divisor (GCD). Working with our previous result:

• \( \frac{4}{400} \),

we need to identify the GCD of 4 and 400, which is 4. We then divide both the top and bottom by this number:

• \( \frac{4 \div 4}{400 \div 4} = \frac{1}{100} \).

By simplifying fractions, we make them easier to read and interpret. It also allows for more straightforward arithmetic in future calculations. Simplification is an essential technique, especially when dealing with large numbers or multiple terms, and reflects a deeper understanding of fractions and their properties. Knowing how and when to simplify is key in ensuring clarity and accuracy in mathematics.