Problem 72 - Algebra for College Students

Question

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(\frac{-48 a b^{2}}{-6 a^{3} b^{5}}\right)^{-2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{64}{a^4 b^6}\).

Step 1: Simplify the Expression Inside the Parentheses

First, simplify the rational expression \[\frac{-48 a b^2}{-6 a^3 b^5}\]by canceling common factors in the numerator and denominator. The negative sign in both the numerator and denominator will cancel out. Divide both numbers and variables:1. \(-48\) divided by \(-6\) is \(8\).2. For \(a\), \(a\) in the numerator and \(a^3\) in the denominator gives \(a^{1-3} = a^{-2}\).3. For \(b\), \(b^2\) in the numerator and \(b^5\) in the denominator gives \(b^{2-5} = b^{-3}\).So, the simplified expression is \[8 a^{-2} b^{-3}.\]

Step 2: Apply the Negative Exponent Rule

Now apply the negative exponent rule, which states that \[(x^{-n}) = \frac{1}{x^n}.\]This rule gives\[(8 a^{-2} b^{-3})^{-2} = \frac{1}{(8 a^{-2} b^{-3})^2}.\]

Step 3: Distribute the Exponent to Each Factor

Distribute the exponent \(-2\) to each factor in the expression. In the expression \((8 a^{-2} b^{-3})^2\), apply the power to each factor:1. \((8)^2 = 64\)2. \((a^{-2})^2 = a^{-4}\)3. \((b^{-3})^2 = b^{-6}\)This results in:\[64 a^{-4} b^{-6}.\]

Step 4: Express Using Positive Exponents

To convert to positive exponents, use the rule that states \(x^{-n} = \frac{1}{x^n}\):Convert each factor with a negative exponent:1. \(a^{-4}\) becomes \(\frac{1}{a^4}\).2. \(b^{-6}\) becomes \(\frac{1}{b^6}\).Thus the expression becomes\[\frac{64}{a^4 b^6}.\]

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is like cleaning up a messy room. You aim to make everything as neat and simple as possible. By simplifying, we reduce expressions to their most basic form. This is important because it makes them easier to work with.
In our exercise, the first step was to simplify the fraction \( \frac{-48 a b^2}{-6 a^3 b^5} \). You do this by canceling out common factors both in the numerator and the denominator. Recognizing that both \(-48\) and \(-6\) share a common factor of \(-6\) makes simplifying straightforward. We divide \(-48 \div -6\) to give \(8\).

• For variables, we subtract the powers of the common bases, as in \(a\): here, it changes from \(a^{1}\) to \(a^{1-3} = a^{-2}\).
• Similarly for \(b\), it changes from \(b^2\) to \(b^{2-5} = b^{-3}\).

The simplified expression \(8 a^{-2} b^{-3}\) is then ready for further transformation.

Negative Exponents

Negative exponents might seem tricky at first, but they are quite logical. A negative exponent essentially indicates the reciprocal of the base raised to the opposite positive exponent. For instance, \(x^{-n} = \frac{1}{x^n}\).
When we encounter \((8 a^{-2} b^{-3})^{-2}\), we're dealing with a compound negative exponent situation. First, we address the outer \(-2\) exponent:

• Applying it to \(8\) gives \((8)^2 = 64\).
• Next, for \(a^{-2}\), \((a^{-2})^2 = a^{-4}\).
• And for \(b^{-3}\), \((b^{-3})^2 = b^{-6}\).

The expression becomes \(64 a^{-4} b^{-6}\). These steps follow the rule of distributing exponent to each factor inside the parenthesis.

Rational Expressions

Rational expressions involve the division of one polynomial by another. They are fundamentally fractions where both the numerator and the denominator are expressions rather than simple numbers. Simplifying and dealing with rational expressions requires understanding exponent rules and factoring.
In our problem, we had a rational expression within the parentheses \(\left( \frac{-48 a b^2}{-6 a^3 b^5} \right)\). The completion of simplification produced the rational expression \(\frac{64}{a^4 b^6}\), requiring the conversion of negative to positive exponents.

• To express \(a^{-4}\) as a positive exponent, we remember the rule: change the location in a fraction. Thus, it becomes \(\frac{1}{a^4}\).
• For \(b^{-6}\), similarly, it becomes \(\frac{1}{b^6}\).

Finally, we obtained \(\frac{64}{a^4 b^6}\) which is the simplified, positively-exponentiated rational expression.