Operating with Rational Expressions - Algebra 2
Card 1 of 30
What is the result of adding $\frac{a}{b}+\frac{c}{b}$ when denominators match?
What is the result of adding $\frac{a}{b}+\frac{c}{b}$ when denominators match?
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$\frac{a+c}{b}$. Combine numerators over the common denominator.
$\frac{a+c}{b}$. Combine numerators over the common denominator.
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What additional restriction occurs when dividing by $\frac{x-2}{x+1}$?
What additional restriction occurs when dividing by $\frac{x-2}{x+1}$?
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$\frac{x-2}{x+1}\neq 0$, so $x\neq 2$ (and also $x\neq -1$). The divisor cannot equal zero for division to be defined.
$\frac{x-2}{x+1}\neq 0$, so $x\neq 2$ (and also $x\neq -1$). The divisor cannot equal zero for division to be defined.
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What is $\frac{x-3}{x}\div\frac{x-3}{x+1}$ simplified (as an expression)?
What is $\frac{x-3}{x}\div\frac{x-3}{x+1}$ simplified (as an expression)?
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$\frac{x+1}{x}$. Cancel common factor $(x-3)$ after multiplying by reciprocal.
$\frac{x+1}{x}$. Cancel common factor $(x-3)$ after multiplying by reciprocal.
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What is $\frac{1}{x}\div\frac{1}{x+5}$ simplified?
What is $\frac{1}{x}\div\frac{1}{x+5}$ simplified?
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$\frac{x+5}{x}$. Multiply by reciprocal: $\frac{1}{x} \cdot \frac{x+5}{1}$.
$\frac{x+5}{x}$. Multiply by reciprocal: $\frac{1}{x} \cdot \frac{x+5}{1}$.
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What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?
What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?
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$1$. Rewrite division as multiplication: $\frac{x^2-1}{x+1} \cdot \frac{1}{x-1}$.
$1$. Rewrite division as multiplication: $\frac{x^2-1}{x+1} \cdot \frac{1}{x-1}$.
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What is $\frac{x}{x-1}\div\frac{x+2}{x-1}$ simplified?
What is $\frac{x}{x-1}\div\frac{x+2}{x-1}$ simplified?
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$\frac{x}{x+2}$. Multiply by reciprocal and cancel $(x-1)$.
$\frac{x}{x+2}$. Multiply by reciprocal and cancel $(x-1)$.
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What is $\frac{2}{x}\div\frac{3}{x+1}$ simplified?
What is $\frac{2}{x}\div\frac{3}{x+1}$ simplified?
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$\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x} \cdot \frac{x+1}{3}$.
$\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x} \cdot \frac{x+1}{3}$.
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What is $\frac{1}{x+1}-\frac{1}{x-1}$ simplified?
What is $\frac{1}{x+1}-\frac{1}{x-1}$ simplified?
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$\frac{-2}{(x+1)(x-1)}$. Use LCD $(x+1)(x-1)$ and subtract: $(x-1-x-1)$.
$\frac{-2}{(x+1)(x-1)}$. Use LCD $(x+1)(x-1)$ and subtract: $(x-1-x-1)$.
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What value must be excluded when simplifying $\frac{(x-3)(x+1)}{x-3}$?
What value must be excluded when simplifying $\frac{(x-3)(x+1)}{x-3}$?
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$x\neq 3$. The factor $(x-3)$ in original expression cannot equal zero.
$x\neq 3$. The factor $(x-3)$ in original expression cannot equal zero.
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What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?
What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?
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$\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.
$\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.
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What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?
What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?
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$\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.
$\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.
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What does it mean for rational expressions to be closed under division?
What does it mean for rational expressions to be closed under division?
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The quotient is rational if dividing by a nonzero rational expression. Division by zero is excluded, but the result remains rational.
The quotient is rational if dividing by a nonzero rational expression. Division by zero is excluded, but the result remains rational.
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What is $\frac{2}{x+1}+\frac{3}{x-1}$ simplified?
What is $\frac{2}{x+1}+\frac{3}{x-1}$ simplified?
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$\frac{5x+1}{(x+1)(x-1)}$. Use LCD and add: $\frac{2(x-1)+3(x+1)}{(x+1)(x-1)}$.
$\frac{5x+1}{(x+1)(x-1)}$. Use LCD and add: $\frac{2(x-1)+3(x+1)}{(x+1)(x-1)}$.
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What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?
What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?
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$\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.
$\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.
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What is $\frac{1}{x+2}-\frac{1}{x}$ simplified?
What is $\frac{1}{x+2}-\frac{1}{x}$ simplified?
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$\frac{-2}{x(x+2)}$. Find LCD $x(x+2)$ and subtract: $\frac{x-(x+2)}{x(x+2)}$.
$\frac{-2}{x(x+2)}$. Find LCD $x(x+2)$ and subtract: $\frac{x-(x+2)}{x(x+2)}$.
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What is $\frac{1}{x^2-1}-\frac{1}{x+1}$ simplified?
What is $\frac{1}{x^2-1}-\frac{1}{x+1}$ simplified?
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$\frac{-x}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{1-(x-1)}{(x-1)(x+1)}$.
$\frac{-x}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{1-(x-1)}{(x-1)(x+1)}$.
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What is $\frac{x^2-9}{x-3}\cdot\frac{1}{x+3}$ simplified?
What is $\frac{x^2-9}{x-3}\cdot\frac{1}{x+3}$ simplified?
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$1$. Factor $x^2-9=(x+3)(x-3)$ and cancel both factors.
$1$. Factor $x^2-9=(x+3)(x-3)$ and cancel both factors.
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What does it mean for rational expressions to be closed under multiplication?
What does it mean for rational expressions to be closed under multiplication?
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The product of rational expressions is rational. The result of multiplying rational expressions is rational.
The product of rational expressions is rational. The result of multiplying rational expressions is rational.
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What is $\frac{x}{x^2-1}+\frac{1}{x+1}$ simplified?
What is $\frac{x}{x^2-1}+\frac{1}{x+1}$ simplified?
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$\frac{2x+1}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{x+(x-1)}{(x-1)(x+1)}$.
$\frac{2x+1}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{x+(x-1)}{(x-1)(x+1)}$.
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What is the rule for dividing by a rational expression $\frac{P(x)}{Q(x)}$?
What is the rule for dividing by a rational expression $\frac{P(x)}{Q(x)}$?
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Multiply by the reciprocal $\frac{Q(x)}{P(x)}$, with $P(x)\neq 0$. Division by a fraction means multiplying by its reciprocal.
Multiply by the reciprocal $\frac{Q(x)}{P(x)}$, with $P(x)\neq 0$. Division by a fraction means multiplying by its reciprocal.
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What is $\frac{1}{x}-\frac{1}{x+1}$ simplified?
What is $\frac{1}{x}-\frac{1}{x+1}$ simplified?
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$\frac{1}{x(x+1)}$. Find LCD $x(x+1)$ and subtract: $\frac{x+1-x}{x(x+1)}$.
$\frac{1}{x(x+1)}$. Find LCD $x(x+1)$ and subtract: $\frac{x+1-x}{x(x+1)}$.
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What is the result of subtracting $\frac{a}{b}-\frac{c}{b}$ when denominators match?
What is the result of subtracting $\frac{a}{b}-\frac{c}{b}$ when denominators match?
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$\frac{a-c}{b}$. Subtract numerators over the common denominator.
$\frac{a-c}{b}$. Subtract numerators over the common denominator.
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What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?
What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?
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$\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.
$\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.
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What is $\frac{x}{x^2}\cdot\frac{x^3}{x}$ simplified (for $x\neq 0$)?
What is $\frac{x}{x^2}\cdot\frac{x^3}{x}$ simplified (for $x\neq 0$)?
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$x$. Simplify powers: $\frac{x \cdot x^3}{x^2 \cdot x} = \frac{x^4}{x^3} = x$.
$x$. Simplify powers: $\frac{x \cdot x^3}{x^2 \cdot x} = \frac{x^4}{x^3} = x$.
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What is $\frac{x^2-1}{x+1}\cdot\frac{1}{x-1}$ simplified?
What is $\frac{x^2-1}{x+1}\cdot\frac{1}{x-1}$ simplified?
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$1$. Factor $x^2-1=(x+1)(x-1)$ and cancel both factors.
$1$. Factor $x^2-1=(x+1)(x-1)$ and cancel both factors.
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What is $\frac{x}{x-1}\cdot\frac{x-1}{x+2}$ simplified?
What is $\frac{x}{x-1}\cdot\frac{x-1}{x+2}$ simplified?
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$\frac{x}{x+2}$. Cancel the common factor $(x-1)$.
$\frac{x}{x+2}$. Cancel the common factor $(x-1)$.
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What is $\frac{2}{x}\cdot\frac{3}{x+1}$ simplified?
What is $\frac{2}{x}\cdot\frac{3}{x+1}$ simplified?
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$\frac{6}{x(x+1)}$. Multiply straight across: $(2\cdot 3)/(x(x+1))$.
$\frac{6}{x(x+1)}$. Multiply straight across: $(2\cdot 3)/(x(x+1))$.
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What is the general multiplication rule for $\frac{a}{b}\cdot\frac{c}{d}$?
What is the general multiplication rule for $\frac{a}{b}\cdot\frac{c}{d}$?
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$\frac{ac}{bd}$ (with $b\neq 0$ and $d\neq 0$). Multiply numerators and denominators separately.
$\frac{ac}{bd}$ (with $b\neq 0$ and $d\neq 0$). Multiply numerators and denominators separately.
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What is $\frac{x}{x+2}+\frac{2}{x+2}$ simplified?
What is $\frac{x}{x+2}+\frac{2}{x+2}$ simplified?
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$\frac{x+2}{x+2}=1$. Add numerators over common denominator: $(x+2)/(x+2)$.
$\frac{x+2}{x+2}=1$. Add numerators over common denominator: $(x+2)/(x+2)$.
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What is $\frac{x}{x^2}+\frac{1}{x}$ simplified (for $x\neq 0$)?
What is $\frac{x}{x^2}+\frac{1}{x}$ simplified (for $x\neq 0$)?
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$\frac{2}{x}$. Rewrite as $\frac{1}{x}+\frac{1}{x}=\frac{2}{x}$.
$\frac{2}{x}$. Rewrite as $\frac{1}{x}+\frac{1}{x}=\frac{2}{x}$.
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