1Cademy - Evaluating $$\frac{a^2 + 2ab + b^2}{3ab^3}$$ for Given Values of $$a$$ and $$b$$

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Evaluating $\frac{x^2+8x+7}{x^2-4}$ for $x = 0$ , $x = 2$ , and $x = -1$

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Evaluating $\frac{a^2 + 2ab + b^2}{3ab^3}$ for Given Values of $a$ and $b$

Evaluate the two-variable rational expression $\frac{a^2 + 2ab + b^2}{3ab^3}$ for three pairs of values by substituting both variables simultaneously and simplifying the numerator and denominator independently before reducing.

ⓐ When $a = 1$ and $b = 2$ : Replace $a$ with $1$ and $b$ with $2$ to get $\frac{(1)^2 + 2(1)(2) + (2)^2}{3(1)(2)^3}$ . Simplify the numerator: $1 + 4 + 4 = 9$ . Simplify the denominator: $3(1)(8) = 24$ . Reduce by dividing numerator and denominator by their GCF of $3$ : $\frac{9}{24} = \frac{3}{8}$ .

ⓑ When $a = -2$ and $b = -1$ : Replace $a$ with $-2$ and $b$ with $-1$ to get $\frac{(-2)^2 + 2(-2)(-1) + (-1)^2}{3(-2)(-1)^3}$ . Simplify the numerator: $4 + 4 + 1 = 9$ . Simplify the denominator: $3(-2)(-1) = 6$ . Reduce: $\frac{9}{6} = \frac{3}{2}$ .

ⓒ When $a = \frac{1}{3}$ and $b = 0$ : Replace $a$ with $\frac{1}{3}$ and $b$ with $0$ to get $\frac{\left(\frac{1}{3}\right)^2 + 2\left(\frac{1}{3}\right)(0) + (0)^2}{3\left(\frac{1}{3}\right)(0)^3}$ . Simplify the numerator: $\frac{1}{9} + 0 + 0 = \frac{1}{9}$ . Simplify the denominator: $3 \cdot \frac{1}{3} \cdot 0 = 0$ . Because the denominator equals zero, the expression is undefined.

This example extends the evaluation of rational expressions from one variable to two. Parts ⓐ and ⓑ require reducing the resulting fraction to lowest terms — dividing both $9$ and $24$ by $3$ in part ⓐ, and both $9$ and $6$ by $3$ in part ⓑ. When both variables are negative (part ⓑ), careful application of sign rules for exponents and multiplication is essential — for instance, $(-1)^3 = -1$ , causing the two negatives in the denominator to produce a positive result. Part ⓒ demonstrates that when a substitution makes the denominator zero — here $b = 0$ zeroes out the $b^3$ factor — the expression is undefined, regardless of the numerator's value.

Updated 2026-04-21

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OpenStax Elementary Algebra

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