Problem 31 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(\frac{2^{-1}}{3^{-2}}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{81}{4} \).

Step 1: Simplify the Fraction

Begin simplifying the expression inside the parenthesis, which is \( \frac{2^{-1}}{3^{-2}} \). Remember that \( a^{-b} = \frac{1}{a^b} \). This means \( 2^{-1} = \frac{1}{2} \) and \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \). So, the fraction becomes: \( \frac{\frac{1}{2}}{\frac{1}{9}} \).

Step 2: Divide the Fractions

To divide fractions, multiply by the reciprocal. Thus, \( \frac{\frac{1}{2}}{\frac{1}{9}} \) becomes \( \frac{1}{2} \times \frac{9}{1} \). This simplifies to \( \frac{9}{2} \).

Step 3: Apply the Exponent

Now apply the square to the fraction: \( \left( \frac{9}{2} \right)^2 \). Square both the numerator and the denominator separately: \( \frac{9^2}{2^2} \).

Step 4: Calculate the Squares

Calculate \( 9^2 \) and \( 2^2 \): \( 9^2 = 81 \) and \( 2^2 = 4 \). So, \( \left( \frac{9}{2} \right)^2 = \frac{81}{4} \).

Key Concepts

Negative Exponents

When dealing with negative exponents, many students find them a bit tricky. An exponent represents how many times to multiply the number by itself. A negative exponent, however, means that the number should be divided, rather than multiplied. Essentially, a negative exponent indicates reciprocal action. For instance, with a base of 2 and a negative exponent of -1, we express this as:

• \( 2^{-1} = \frac{1}{2^1} = \frac{1}{2} \)

Likewise, for 3 with a negative exponent of -2, it becomes:

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

These transformations are crucial for simplifying expressions correctly.

Fraction Simplification

Once we convert negative exponents into fractions, the next step is simplification. In our example, we have fractions inside a larger fraction, like this:

• \( \frac{\frac{1}{2}}{\frac{1}{9}} \)

To simplify such an expression, we use the concept of multiplying by the reciprocal. This means flipping the denominator and multiplying, transforming our problem into:

• \( \frac{1}{2} \times \frac{9}{1} \)
• Which simplifies neatly to \( \frac{9}{2} \)

This step is about changing division into multiplication, which often simplifies the calculations significantly.

Exponentiation

Exponentiation is the process of raising numbers to certain powers. In our context, after simplifying the fraction, we need to apply an exponent to the entire fraction, specifically squaring it. This means the fraction will be multiplied by itself:

• \( \left( \frac{9}{2} \right)^2 \)

For squaring a fraction, square both the numerator and the denominator individually:

• \( 9^2 = 81 \)
• \( 2^2 = 4 \)
• Resulting in \( \frac{81}{4} \)

This method ensures that our work is consistent with exponential rules.

Mathematical Operations

Mathematical operations encompass all the actions needed to simplify, solve, or manipulate expressions using various techniques. Within our problem, we use several: converting negative exponents, simplifying fractions, and applying exponents. Here are the key steps summarized:

• Convert negative exponents to fractions.
• Simplify complex fractions by multiplication.
• Apply further exponentiation as needed.

Understanding these processes in sequence helps develop a systematic approach toward tackling complex mathematical expressions. With practice, these operations become intuitive and quick to perform.