Complex Rational Expressions

Complex rational expressions. In MathBound, the free Algebra I adventure game, Double-Decker Bot asks these problems, and the only way past is to solve them.

Part of More topics: taught by some textbooks beyond the usual course.

Play this battle free

10 questions, no sign-up, nothing saved. Climbs from the basics to harder questions.

What you will practice

Simplify by Division (Method 1). Simplify a complex fraction by combining the numerator and denominator into single fractions first then multiplying by the reciprocal.
Simplify by LCD (Method 2). Simplify a complex fraction by multiplying the main numerator and denominator by the LCD of all minor monomial fractions.
Simplify Negative Exponents. Rewrite expressions with negative exponents as complex fractions before simplifying.

Worked examples

Made by the game’s own question engine. Try each one, then open its steps.

Example 1 · Simplify by Division (Method 1)

Simplify:

\frac{2 - \frac{1}{2}}{2 + \frac{1}{4}}

$\frac{3}{2}$
$2$
$\frac{2}{3}$
$\frac{2}{9}$

Show the worked steps

$Top:$ $2 - \frac{1}{2} = \frac{3}{2} .$ $Bottom:$ $2 + \frac{1}{4} = \frac{9}{4} .$
$\frac{3}{2} \cdot \frac{4}{9} = \frac{2}{3} .$

Answer: $\frac{2}{3}$

Example 2 · Simplify by LCD (Method 2)

To

simplify

\frac{\frac{1}{2 x ^{2}} + \frac{2}{x ^{2} y}}{\frac{2}{2 x ^{2} y} - \frac{2}{6 x ^{2} y}}

by

the

LCD

method,

multiply

its

numerator

and

denominator

by

what?

$6 x^{2} y$
$24 x^{8} y^{3}$
$6 x y$
$x^{2} y$

Show the worked steps

$Numbers:$ $the$ $LCM$ $of$ $2, 1, 2, 6$ $is$ $6 .$
$Letters:$ $take$ $each$ $to$ $its$ $highest$ $power,$ $giving$ $6 x^{2} y .$

Answer: $6 x^{2} y$

Example 3 · Simplify Negative Exponents

Simplify:

(2 x^{- 1} + 3 y^{- 1})^{- 1}

$\frac{2 y + 3 x}{x y}$
$\frac{x y}{2 x + 3 y}$
$\frac{x y}{2 y + 3 x}$
$2 x + 3 y$

Show the worked steps

$Inside:$ $\frac{2}{x} + \frac{3}{y} = \frac{2 y + 3 x}{x y} .$
$The$ $outer$ $- 1$ $power$ $flips$ $it:$ $\frac{x y}{2 y + 3 x} .$

Answer: $\frac{x y}{2 y + 3 x}$

Where it is in your textbook

Textbook	Lessons
Glencoe Algebra 1	11-7
OpenStax Elementary Algebra 2e	8.5

Ready for Double-Decker Bot?

Play this battle free Play the whole adventure