Problem 18 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 1. $$ \frac{b^{65}}{27 a^{2}} \cdot \frac{18 a^{41}}{5 b^{90}} $$

Step-by-Step Solution

Answer

The simplified product is \( \frac{2 a^{39}}{15 b^{25}} \).

Step 1: Understand the Problem

We need to multiply two fractions together: \( \frac{b^{65}}{27 a^{2}} \) and \( \frac{18 a^{41}}{5 b^{90}} \). Each fraction has a numerator and a denominator that must be multiplied together.

Step 2: Multiply the Numerators

The numerators are \( b^{65} \) and \( 18 a^{41} \). To multiply them, combine their terms: \( 18 a^{41} imes b^{65} = 18 a^{41} b^{65} \).

Step 3: Multiply the Denominators

The denominators are \( 27 a^{2} \) and \( 5 b^{90} \). Multiply these together as well: \( 27 a^{2} imes 5 b^{90} = 135 a^{2} b^{90} \).

Step 4: Write the Product of the Fractions

Now that we have the products for numerator and denominator, write the new fraction: \( \frac{18 a^{41} b^{65}}{135 a^{2} b^{90}} \).

Step 5: Simplify the Fraction Numerically

We simplify the fraction \( \frac{18}{135} \) by finding the greatest common divisor (GCD) of 18 and 135, which is 9. Thus, the simplified form is \( \frac{18 \div 9}{135 \div 9} = \frac{2}{15} \).

Step 6: Simplify the Variable Part of the Fraction

Simplify the fraction by reducing the powers of \( a \) and \( b \): \( a^{41-2} = a^{39} \) and \( b^{65-90} = b^{-25} \). Thus, the fraction becomes \( \frac{2 a^{39}}{15 b^{25}} \).

Step 7: Final Answer

The fraction \( \frac{2 a^{39}}{15 b^{25}} \) is in its simplest form. There are no further common factors to reduce.

Key Concepts

Multiplying Fractions

Multiplying fractions involves a straightforward process that requires us to multiply numerators and denominators separately. For the given exercise, we had two fractions:

• \( \frac{b^{65}}{27 a^{2}} \)
• \( \frac{18 a^{41}}{5 b^{90}} \)

The numerators \((b^{65}\) and \(18a^{41})\) were multiplied to get \(18a^{41}b^{65}\). Similarly, the denominators \((27a^{2}\) and \(5b^{90})\) were multiplied to yield \(135a^{2}b^{90}\).
This process results in a single fraction, \( \frac{18a^{41}b^{65}}{135a^{2}b^{90}} \). The key is to treat the fractions like any multiplication problem, focusing on how we multiply terms systematically. Remember, when multiplying fractions, always multiply straight across: numerators with numerators and denominators with denominators.

Power Rules in Algebra

The power rules help us simplify terms with exponents when multiplying or dividing them. In our exercise, we had to deal with powers of variables. The rule we applied states:

• \( x^{a} \times x^{b} = x^{a+b} \) when multiplying like bases
• \( x^{a} \div x^{b} = x^{a-b} \) when dividing like bases

Essentially, when you're multiplying terms with the same base, such as \(a^{41}\) and \(a^{2}\), you add the exponents. Conversely, for division, as in \(b^{65}\) divided by \(b^{90}\), you subtract the exponents. Thus, applying these rules, we simplified \(a^{41}\) and \(a^{2}\) to \(a^{39}\), and \(b^{65}\) and \(b^{90}\) to \(b^{-25}\), which became \(\frac{1}{b^{25}}\) since exponent \(-25\) implies the reciprocal.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial for simplifying fractions numerically. When given a numerical fraction like \(\frac{18}{135}\), we need to determine the largest number that divides both the numerator and the denominator.
For 18 and 135, their GCD is 9. Here's how you can easily find it:

• List the factors of both numbers.
• Identify the largest common factor.

Once identified, divide both the numerator and the denominator by the GCD to simplify the fraction:
\[\frac{18}{135} = \frac{18 \div 9}{135 \div 9} = \frac{2}{15}\]Finding the GCD ensures that the fraction is reduced to its simplest form, making it easier to understand and work with.