Problem 17 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 1. $$ \frac{12 x^{61}}{7 y^{15}} \cdot \frac{y}{8 x^{27}} $$

Step-by-Step Solution

Answer

\(\frac{3 x^{34}}{14 y^{14}}\)

Step 1: Multiply the Numerators

To begin, multiply the numerators of the fractions. The numerators are: \(12 x^{61}\) and \(y\).Multiplying these together gives:\(12 x^{61} imes y = 12 x^{61} y\).

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions.The denominators are: \(7 y^{15}\) and \(8 x^{27}\).Multiplying these together gives:\(7 y^{15} imes 8 x^{27} = 56 x^{27} y^{15}\).

Step 3: Combine the Products into a Single Fraction

Combine the result of Step 1 and Step 2 into a single fraction:\[\frac{12 x^{61} y}{56 x^{27} y^{15}}\].

Step 4: Simplify the Fraction - Coefficients

First, simplify the coefficients (the numbers). Divide the coefficients 12 and 56 by their greatest common divisor (GCD), which is 4:\(\frac{12}{56} = \frac{3}{14}\).

Step 5: Simplify the Fraction - Variables in the Numerator

Now, perform the simplification for each variable.Starting with \(x\):We have \(x^{61}\) in the numerator and \(x^{27}\) in the denominator:Apply the quotient rule for exponents: \(x^{61} / x^{27} = x^{61-27} = x^{34}\).

Step 6: Simplify the Fraction - Variables in the Denominator

For \(y\):We have \(y\) in the numerator and \(y^{15}\) in the denominator:Apply the quotient rule for exponents: \(y^1 / y^{15} = y^{1-15} = y^{-14}\). Since we don't want negative exponents in the final answer, we rewrite:\(y^{-14} = \frac{1}{y^{14}}\).

Step 7: Final Simplification of the Fraction

Combine the simplified components to form the final fraction:\[\frac{3 x^{34}}{14 y^{14}}\]. This is the fully simplified form.

Key Concepts

Fraction Simplification

Fraction simplification is all about making a fraction as simple as possible, while keeping its value the same. Simplifying fractions means reducing them to their smallest form. When you multiply fractions, like in the original exercise, it's a good idea to simplify both the numerators and denominators.

For example, let's say you have a fraction \( \frac{12}{56} \). To simplify it, you need to find the greatest common divisor (GCD) of both numbers. The GCD is the largest number that divides both numbers evenly. Here, the GCD of 12 and 56 is 4. So, you divide both by 4:

• 12 ÷ 4 = 3
• 56 ÷ 4 = 14

Thus, \( \frac{12}{56} \) simplifies to \( \frac{3}{14} \).

This shows how simplifying fractions is essentially about finding and using the GCD to make the fraction smaller. When fractions get simpler, they become easier to work with in both calculations and real-life situations.

Rational Expressions

Rational expressions are similar to fractions, but they involve polynomials, which are mathematical expressions involving variables. When multiplying two rational expressions, you need to multiply both the numerators and denominators separately.

Take the original expression \( \frac{12 x^{61}}{7 y^{15}} \cdot \frac{y}{8 x^{27}} \). You begin by multiplying the numerators: \(12x^{61} \times y\), and then the denominators: \( 7y^{15} \times 8x^{27} \). This forms a new fraction where you can simplify both the numbers and variables.

Simplifying rational expressions doesn't just stop at the numbers. Variables with exponents can also be simplified using the rules of exponents. When you divide like terms, you subtract the exponents. So, for the \(x\) variable: \(x^{61} \div x^{27} \), you subtract 27 from 61, giving \(x^{34}\). The same applies to the \(y\) variable: \(y^{1} \div y^{15} \), resulting in \(y^{-14}\), which changes the position to the denominator as \( \frac{1}{y^{14}} \).

Understanding rational expressions requires combining your knowledge of fraction multiplication with exponent rules, making it a versatile tool for algebraic problem-solving.

Coefficients Simplification

Coefficients in algebra are the numerical part of the terms involving variables. When working with rational expressions or fractions, simplifying coefficients is a crucial step.

In the given exercise, the coefficients in the fraction \( \frac{12x^{61}}{56x^{27}y^{15}} \) need to be simplified. You do this by identifying the greatest common divisor (GCD) of 12 and 56, which is 4. By dividing each coefficient by 4, you get:

• 12 ÷ 4 = 3
• 56 ÷ 4 = 14

This simplification results in a cleaner fraction, \( \frac{3x^{34}}{14y^{14}} \), which shows how simplification can make calculations more straightforward.

Whether you're simplifying coefficients, constants, or variables, keeping expressions neat and manageable is key. This not only helps solve algebraic problems efficiently but also makes complex expressions easier to understand.