Problem 86 - Algebra for College Students

Question

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{3 a+1}}{y^{2 b-3}} \cdot \frac{y^{3 b+4}}{x^{2 a-1}}$$

Step-by-Step Solution

Answer

The simplified form is \(x^{a+2} \cdot y^{b+7}\).

Step 1: Write the expression as a single fraction

Combine the two fractions into a single fraction: \ \[\frac{x^{3a+1}}{y^{2b-3}} \cdot \frac{y^{3b+4}}{x^{2a-1}} = \frac{x^{3a+1} \cdot y^{3b+4}}{y^{2b-3} \cdot x^{2a-1}}\]

Step 2: Apply the properties of exponents

Simplify the expression by combining the exponents. Use the property \(a^m \cdot a^n = a^{m+n}\) for both the numerator and the denominator: \ \[\frac{x^{3a+1} \cdot y^{3b+4}}{y^{2b-3} \cdot x^{2a-1}} = \frac{x^{(3a+1)} \cdot y^{(3b+4)}}{x^{(2a-1)} \cdot y^{(2b-3)}}\]

Step 3: Simplify the exponents

Subtract the exponents of like bases in the numerator and the denominator, using the property \((a^m)/(a^n) = a^{m-n}\): \ \[x^{(3a+1)-(2a-1)} \cdot y^{(3b+4)-(2b-3)} = x^{(3a+1-2a+1)} \cdot y^{(3b+4-2b+3)} = x^{a+2} \cdot y^{b+7}\]

Key Concepts

Combining Exponents

When dealing with exponents, one essential property is that we can combine them to simplify expressions. This is particularly useful when you multiply or divide terms with the same base. In our given exercise, we have the expression: $$\frac{x^{3a+1}}{y^{2b-3}} \cdot \frac{y^{3b+4}}{x^{2a-1}}$$ To combine these fractions, we first write them as a single fraction: \begin{align*} \frac{x^{3a+1}}{y^{2b-3}} \cdot \frac{y^{3b+4}}{x^{2a-1}} &= \frac{x^{3a+1} \cdot y^{3b+4}}{y^{2b-3} \cdot x^{2a-1}} \ \end{align*} By combining the fractions, we get a single fraction where each base appears with all its exponents added together. In this example, we combined the fractions based on the property of exponents where: - For multiplication: \(a^m \cdot a^n = a^{m+n}\) - For division: \(\frac{a^m}{a^n} = a^{m-n}\) We can now move to the next step and simplify the exponents themselves.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves breaking down the expression to its simplest form. After combining the fractions, our expression looks like this: \[\frac{x^{3a+1} \cdot y^{3b+4}}{y^{2b-3} \cdot x^{2a-1}}\] To simplify, we use the properties of exponents. For the numerator and the denominator, we use: - When bases are multiplied: \(a^m \cdot a^n = a^{m+n}\) - When bases are the same: \(a^m \cdot a^n = a^{m+n}\) In our fraction, we simplify the exponents by combining: \[\frac{x^{(3a+1)} \cdot y^{(3b+4)}}{x^{(2a-1)} \cdot y^{(2b-3)}}\] Next, we subtract exponents for the same bases in the numerator from those in the denominator using \(a^m / a^n = a^{m-n}\). This simplifies the expression further.

Fractional Exponents

Lastly, let's delve into fractional exponents, which we encounter when we simplify further. To simplify the exponents derived earlier: \[x^{(3a+1)-(2a-1)} \cdot y^{(3b+4)-(2b-3)} = x^{(3a+1-2a+1)} \cdot y^{(3b+4-2b+3)} = x^{a+2} \cdot y^{b+7}\] Here’s what’s key: when subtracting, ensure you manage the signs correctly. Remember: - For exponents: \(a^{-m} = \frac{1}{a^m}\) - To handle positive and negative results correctly. This leads to our simplified expression with integer exponents: \(x^{a+2} \cdot y^{b+7}\). Understanding fractional exponents helps us break complex terms so the expressions are more digestible.