Problem 63 - Algebra for College Students

Question

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{4^{-1}-9^{-1}}{2^{-1}+3^{-1}}$$

Step-by-Step Solution

Answer

The fraction is \( \frac{1}{6} \) and the decimal is 0.1667.

Step 1 - Evaluate the Powers

First, compute each term in the numerator and the denominator. Compute the value of \(4^{-1} \), \(9^{-1}\), \(2^{-1} \) and \(3^{-1} \). \(4^{-1} = \frac{1}{4} = 0.25\), \(9^{-1} = \frac{1}{9} \), \(2^{-1} = \frac{1}{2} = 0.5 \), and \(3^{-1} = \frac{1}{3} \).

Step 2 - Simplify the Numerator

Calculate the difference in the numerator: \[ 4^{-1} - 9^{-1} = \frac{1}{4} - \frac{1}{9} \]. Convert the fractions to have a common denominator: \[ \frac{1}{4} = \frac{9}{36}, \ \frac{1}{9} = \frac{4}{36} \]. \[ \frac{9}{36} - \frac{4}{36} = \frac{5}{36} \].

Step 3 - Simplify the Denominator

Calculate the sum in the denominator: \[ 2^{-1} + 3^{-1} = \frac{1}{2} + \frac{1}{3} \]. Convert the fractions to have a common denominator: \[ \frac{1}{2} = \frac{3}{6}, \ \frac{1}{3} = \frac{2}{6} \]. \[ \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \].

Step 4 - Form the Complex Fraction

The complex fraction now is \[ \frac{ \frac{5}{36} }{ \frac{5}{6} } \].

Step 5 - Simplify the Complex Fraction

Simplify the complex fraction by multiplying by the reciprocal: \[ \frac{5}{36} \times \frac{6}{5} = \frac{30}{180} = \frac{1}{6} \].

Step 6 - Evaluate and Round

Convert the simplified fraction into a decimal: \[ \frac{1}{6} = 0.1667 \] when rounded to four decimal places.

Key Concepts

Fractional Powers

Let's begin by understanding what fractional powers are. A fractional power or an exponent represents both a root and a power. For example, the expression \( a^{m/n} \) means you take the \(n\)-th root of \(a\) and then raise the result to the \(m\)-th power. Another example can be seen with \( 4^{-1} \). This means \( \frac{1}{4^{1}} \), which simplifies to \( \frac{1}{4} \). Similarly, \( 9^{-1} \) means \( \frac{1}{9^{1}} \) or \( \frac{1}{9} \).
Understanding fractional powers can help you transform and simplify complex expressions in mathematics. They are a cornerstone for working with equations involving roots and powers.
To compute fractional powers manually, you can break them down as follows:

• Identify the base and the exponent.
• Convert the exponent into a root.
• Perform the root operation followed by any remaining power.

Always remember that when your exponent is negative, like \( a^{-m} \), it means simply taking the reciprocal of \(a^m\).

Common Denominators

To work with fractions effectively, you often need a common denominator. This is a shared multiple of the denominators of the fractions you're working with. For instance, when adding \( \frac{1}{4} \) and \( \frac{1}{9} \), you should find the least common multiple (LCM) of 4 and 9. The LCM of 4 and 9 is 36. Thus, you convert \( \frac{1}{4} \) to \( \frac{9}{36} \) and \( \frac{1}{9} \) to \( \frac{4}{36} \).
Finding common denominators makes it easier to add, subtract, or compare fractions:

• Identify the least common multiple of the denominators.
• Convert each fraction to an equivalent fraction with that common denominator.
• Perform the addition or subtraction by adding/subtracting the numerators and keeping the denominator the same.

This technique is crucial for simplifying complex fractions or solving equations involving fractions.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means you make the numerator and denominator as small as possible while still keeping the same value. For example, \( \frac{30}{180} \) can be simplified. You do this by finding the greatest common divisor (GCD). Both 30 and 180 can be divided by 30:
\( \frac{30}{180} = \frac{30 \div 30}{180 \div 30} = \frac{1}{6} \).
Simplifying fractions helps reduce the complexity of mathematical computations. Here is how to simplify fractions step-by-step:

• Find the GCD of the numerator and the denominator.
• Divide both numerator and denominator by this GCD.

This practice is essential for presenting your answers in the simplest and cleanest form.

Reciprocal

The reciprocal of a number is obtained by swapping its numerator and denominator. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). When a fraction is expressed as \( x \), its reciprocal is \( \frac{1}{x} \). In the final solution step of a complex fraction, you multiply by the reciprocal to simplify:
\( \frac{ \frac{5}{36} }{ \frac{5}{6} } = \frac{5}{36} \times \frac{6}{5} = \frac{30}{180} = \frac{1}{6} \).
Understanding reciprocals is key to simplifying complex fractions and solving rational equations. Here’s how to use reciprocals in calculations:

• Identify the fraction that needs to be inverted.
• Swap its numerator and denominator.
• Multiply the original fraction by this reciprocal to simplify the expression.

Utilizing reciprocals is a powerful tool in fraction manipulation!