Problem 18 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{25}{64} \).

Step 1: Evaluate the First Exponent

Calculate the expression \( \left(\frac{1}{2}\right)^{4} \). This means multiplying \( \frac{1}{2} \) by itself four times: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \). Each multiplication results in \( \frac{1}{4}, \frac{1}{8}, \) and finally \( \frac{1}{16} \). Thus, \( \left(\frac{1}{2}\right)^{4} = \frac{1}{16} \).

Step 2: Evaluate the Second Exponent

Calculate the expression \( \left(\frac{5}{2}\right)^{2} \). This means multiplying \( \frac{5}{2} \) by itself: \( \frac{5}{2} \times \frac{5}{2} = \frac{25}{4} \). Thus, \( \left(\frac{5}{2}\right)^{2} = \frac{25}{4} \).

Step 3: Multiply the Results

Now multiply the results of the two exponents: \( \frac{1}{16} \times \frac{25}{4} \). The multiplication of two fractions involves multiplying the numerators and the denominators separately: \( \frac{1 \times 25}{16 \times 4} = \frac{25}{64} \).

Key Concepts

Fraction Multiplication

Multiplying fractions might seem tricky at first, but it's actually quite straightforward once you get the hang of it. When multiplying two fractions, you simply multiply the numerators together and the denominators together. Let's say you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \). Here is what you do:

• Multiply the numerators: \( a \times c \).
• Multiply the denominators: \( b \times d \).
• Combine them: \( \frac{a \times c}{b \times d} \).

That's all! As an example, if you multiply \( \frac{1}{3} \) by \( \frac{2}{5} \), you multiply the numerators (1 and 2) to get 2, and the denominators (3 and 5) to get 15, resulting in \( \frac{2}{15} \). Remember, fractions can often be simplified after multiplication, if the numerator and denominator have common factors.

Power of a Fraction

Raising a fraction to a power involves repeated multiplication of that fraction. For example, if you have \( \left(\frac{a}{b}\right)^n \), it means you multiply \( \frac{a}{b} \) by itself \( n \) times. Here's how it works in steps:

• Repeat the fraction multiplication step \( n \) times.
• Result in a new fraction: both numerators multiplied for the new numerator, and both denominators multiplied for the new denominator.

For instance, raising \( \left(\frac{1}{2}\right)^4 \) means you multiply \( \frac{1}{2} \) by itself four times: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \). This results in \( \frac{1}{16} \). Simplifying can sometimes make the result easier to interpret, but in this case, \( \frac{1}{16} \) is already in its simplest form. Remember, keep track of both the numerators and denominators as you go through each repeated multiplication.

Evaluating Expressions

When you evaluate mathematical expressions, especially those involving fractions and exponents, you follow specific steps. Here’s a simple checklist to make it easier:

• Address any exponents first. Do this by applying the power of a fraction rule, as we did earlier.
• Once you've calculated the powers, proceed with the multiplication of fractions.
• Simplify the result if possible, looking for any numbers that share common factors in the numerator and denominator.

For the given expression \( \left(\frac{1}{2}\right)^{4} \cdot\left(\frac{5}{2}\right)^{2} \), first we work out each power: \( \left(\frac{1}{2}\right)^{4} = \frac{1}{16} \) and \( \left(\frac{5}{2}\right)^{2} = \frac{25}{4} \). Next, multiply these simplified fractions: \( \frac{1}{16} \times \frac{25}{4} = \frac{25}{64} \). There’s no further simplification needed as \( 25 \) and \( 64 \) don’t have common factors. This step-by-step method ensures you don’t miss any crucial parts of evaluating the expression.