Problem 63 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Use only positive exponents. $$\frac{5 x^{4} y}{3 x y^{2}} \cdot \frac{9 x y}{x^{2} y}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{15x^{2}}{y}\).

Step 1: Multiply the fractions

To begin with, go ahead and simply multiply the two fractions together. To do this, multiply the numerators \(5x^4y\) and \(9xy\) together to get the new numerator. Multiply the denominators \(3xy^2\) and \(x^2y\) together for the new denominator. Thus, our new fraction becomes: \(\frac{45x^{5}y^{2}}{3x^{3}y^{3}}\).

Step 2: Simplify the coefficients

We can simplify the coefficients (45 and 3) in the new fraction obtained in step 1. This will lead to: \(\frac{15 x^{5}y^{2}}{x^{3}y^{3}}\).

Step 3: Apply the laws of exponents and simplify the exponents

The law of exponents states that for any nonzero number \(a\) and integers \(m\) and \(n\), \(\frac{a^m}{a^n} = a^{m-n}\). Therefore, subtract the exponent in the denominator from the exponent in the numerator for both the \(x\) terms and the \(y\) terms. This gives us: \(15x^{5-3}y^{2-3} = 15x^{2}y^{-1}\).

Step 4: Using only positive exponents

The requirement that only positive exponents are to be used means we need to rewrite the \(y^{-1}\) so that the exponent is positive. The identity \(a^{-n} = \frac{1}{a^n}\) can be applied to do this. Hence, we get the final expression as: \(15x^{2}\frac{1}{y} = \frac{15x^{2}}{y}\)

Key Concepts

Laws of Exponents

When working with exponents, there are specific rules, known as the Laws of Exponents, that can help simplify expressions. Understanding these laws is crucial in algebra.
One key law is the quotient of powers rule: \[\frac{a^m}{a^n} = a^{m-n}\]This rule tells us that when dividing two expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
This law was used when simplifying terms like \(x^5/x^3\) in our original expression. By applying the rule, this simplifies to \(x^{5-3} = x^2\).Exploring these rules:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)
• Power of a product: \((ab)^n = a^n \cdot b^n\)

Remember, these rules apply only when the bases are the same, and they offer a systematic way to simplify complex expressions.

Multiplying Fractions

Multiplying fractions is more straightforward than it might seem. The key idea is to multiply numerators together and denominators together separately.
For example, if you have \(\frac{a}{b} \cdot \frac{c}{d}\), the result is \(\frac{ac}{bd}\).In our problem, the initial expression involved multiplying two fractions:\[\frac{5x^4y}{3xy^2} \cdot \frac{9xy}{x^2y}\]The steps involve:

• Numerators: Multiply \(5x^4y\) by \(9xy\) to get \(45x^5y^2\).
• Denominators: Multiply \(3xy^2\) by \(x^2y\) to obtain \(3x^3y^3\).

This results in the new fraction: \(\frac{45x^5y^2}{3x^3y^3}\).Remember, multiplying fractions directly like this simplifies the process, allowing you to tackle the expression systematically.

Positive Exponents

To express a variable with only positive exponents, exceptions like negative powers need to be adjusted. Exponents indicate how many times a number is multiplied by itself.
Negative exponents have special meanings. The expression \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).In the step-by-step solution, the expression \(15x^2y^{-1}\) was problematic because of the negative exponent on \(y\). Using the rule for negative exponents, it was rewritten as:
\(15x^2 \cdot \frac{1}{y} = \frac{15x^2}{y}\).Always aim for expressions with positive exponents for clarity:

• Convert \(a^{-n}\) to \(\frac{1}{a^n}\) to keep exponents positive.
• This approach provides simpler, more intuitive results.

Understanding how to manage negative exponents will lead to cleaner and more comprehensible algebraic expressions.