Chapter 11.1, Problem 52PPE

To determine

Determine whether the statement $\frac{a{b}^3}{b^4}=ab$ is always, sometimes, or never true for real numbers $a\text{and}b$ . Explain.

Answer to Problem 52PPE

Expression $\frac{a{b}^3}{b^4}\ne ab$ is never true.

Explanation of Solution

Given:

The expression: $\frac{a{b}^3}{b^4}=ab$

Concept Used:

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

A rational expression is considered simplified if the numerator and denominator have no factors in common.

Step 1: Factor the numerator and the denominator.

Step 2: List restricted values.

Step 3: Cancel common factors.

Step 4: Simplify and note any restricted values not implied by the expression.

Calculation:

Find the factors of numerator and denominator of the expression $\frac{a{b}^3}{b^4}$ and cancel the common factors from numerator and denominator.

  $\frac{a{b}^3}{b^4}=\frac{a·b·b·b}{b·b·b·b}=\frac{a}{b}$

Simplified Expression $\frac{a{b}^3}{b^4}=\frac{a}{b};\frac{a{b}^3}{b^4}\ne ab$ is never true.

Thus, Expression $\frac{a{b}^3}{b^4}\ne ab$ is never true.