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PSAT Math : Algebraic Fractions

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Example Question #631 : Algebra

Simplify the expression

$(12X^2Y^{-3}Z^2)\left(\frac{X^3Y^4Z^{-3}}{4}\right)$ .

Possible Answers:

$\frac{3X^5Y}{Z}$

$\frac{3}{XYZ}$

$\frac{3Y}{XZ}$

3X^5Z

Correct answer:

$\frac{3X^5Y}{Z}$

Explanation:

We can combine the expression into

$\left(\frac{12X^2X^3Y^{-3}Y^4Z^2Z^{-3}}{4}\right) = \frac{12}{4} X^{(2+3)}Y^{(-3+4)}Z^{(2-3)}$

The combined expression can then be simplified to

$3X^5Y^1Z^{-1} = \frac{3X^5Y}{Z}$

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Example Question #1 : How To Simplify A Fraction

Simplify the expression

$\left(\frac{A^{\frac{3}{2}}\sqrt{B}}{C^{-\frac{1}{2}}}\right)\left(\frac{B^{-\frac{1}{2}}}{2\sqrt{AC}}\right)$ .

Possible Answers:

2A^2

$\frac{A^2}{2}$

$\frac{A}{2}$

A^2

Correct answer:

$\frac{A}{2}$

Explanation:

The expression can be rewritten as

$\left(\frac{A^{\frac{3}{2}}B^\frac{1}{2}}{C^{-\frac{1}{2}}}\right)\left(\frac{B^{-\frac{1}{2}}}{2A^\frac{1}{2}C^\frac{1}{2}}\right)$

We can now move the variables to the numerator and combine alike variables

$(A^\frac{3}{2}B^\frac{1}{2}C^\frac{1}{2})\left(\frac{A^{-\frac{1}{2}}B^{-\frac{1}{2}}C^{-\frac{1}{2}}}{2}\right) = \frac{A^{(\frac{3}{2}-\frac{1}{2})}B^{(\frac{1}{2}-\frac{1}{2})}C^{(\frac{1}{2}-\frac{1}{2})}}{2}$

This becomes

$\frac{A^{\frac{2}{2}}B^0C^0}{2} = \frac{A^1}{2} = \frac{A}{2}$

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Example Question #2 : How To Simplify A Fraction

Simplify the expression

$\frac{a^2-b^2}{a-b}$ .

Possible Answers:

a^2 + b^2

Cannot be simplified

a+b

$\frac{a}{b}-\frac{b}{a}$

Correct answer:

a+b

Explanation:

The numerator of this expression can be factored, giving the expression

$\frac{(a+b)(a-b)}{(a-b)}$

The (a-b) will cancel itself out leaving,

$\frac{a+b\cdot1}{1} = a+b$

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Example Question #3 : How To Simplify A Fraction

Simplify the expression

$\frac{x^2-9}{x+3}$ .

Possible Answers:

x +3

x^2 - 3

x+6

x-3

Correct answer:

x-3

Explanation:

The expression can be factored to the expression

$\frac{(x-3)(x+3)}{(x+3)}$

The $\small (x+3)$ in the numerator and denominator will then cancel each other out to give,

$\small \frac{(x-3)\cdot1}{1} = x -3$

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Example Question #11 : How To Simplify A Fraction

Simplify the expression

$\small \frac{x^2-4x+4}{x-2}$ .

Possible Answers:

$\small x^2$

$\small x-4$

$\small x+2$

$\small x-2$

Correct answer:

$\small x-2$

Explanation:

The expression is factorable into the expression

$\small \frac{(x-2)(x-2)}{(x-2)}$

Cancelling the $\small x-2$ in the numerator and the denominator leaves

$\small x-2$

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Example Question #12 : How To Simplify A Fraction

Simplify the expression,

$\small \frac{a^2+b^2}{a+b}$ .

Possible Answers:

$\small a+b$

$\small \frac{a}{b}+\frac{b}{a}$

Cannot be simplified

$\small a- b$

Correct answer:

Cannot be simplified

Explanation:

The numerator of the expression cannot be factored. Therefore, the denominator cannot divide into the numerator, and the expression is in its simplest form.

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Example Question #13 : How To Simplify A Fraction

Simplify the expression

$\small \frac{x^2+x-6}{x+3}$ .

Possible Answers:

$\small 2x$

$\small x +3$

$\small x - 2$

Cannot be simplified

Correct answer:

$\small x - 2$

Explanation:

The expression's numerator can be factored. Factoring the expression gives,

$\small \frac{(x-2)(x+3)}{(x+3)}$

The numerator and denominator can then both be divided by $\small (x+3)$ to give,

$\small \frac{(x-2)(1)}{(1)} = x-2$

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Example Question #14 : How To Simplify A Fraction

Simplify the expression

$\small \frac{x^3-2x^2+9x-18}{x^2+9}$ .

Possible Answers:

Cannot be simplified

x + 2

$\small x - 3$

$\small x-2$

Correct answer:

$\small x-2$

Explanation:

The expression can be factored to give

$\small \frac{(x-2)(x^2+9)}{(x^2+9)}$ .

$\small (x^2+9)$ can then be divded from the numerator and the denominator to give,

$\small \frac{(x-2)(1)}{(1)} = x -2$

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Example Question #55 : Algebraic Fractions

Simplify the expression

$\small \frac{x^2-7x+10}{x^2-9x+20}$ .

Possible Answers:

$\small 2x$

$\small \frac{x-3}{x-5}$

$\small 2$

$\small \frac{x-2}{x-4}$

Correct answer:

$\small \frac{x-2}{x-4}$

Explanation:

The expression must first be factored in order to be solvable. Both the numerator and the denominator can be factored, which would give

$\small \frac{(x-2)(x-5)}{(x-5)(x-4)}$ .

$\small x-5$ can be divided from both the numerator and the denominator to give

$\small \frac{(x-2)(1)}{(1)(x-4)} = \frac{x-2}{x-4}$

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Example Question #15 : How To Simplify A Fraction

Simplify the expression

$\small \frac{x^3y^{-2}z^2}{xy^3z^2}$ .

Possible Answers:

$\small \frac{x^2z^4}{y}$

$\small \frac{x^2z}{y}$

$\small \frac{x^2}{y}$

$\small \frac{x^2}{y^5}$

Correct answer:

$\small \frac{x^2}{y^5}$

Explanation:

The expression can be rewritten as

$\small x^{(3-1)}y^{(-2-3)}z^{(2-2)} = x^2y^{-5}z^0 = \frac{x^2}{y^5}$

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PSAT Math : Algebraic Fractions

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #631 : Algebra

Example Question #1 : How To Simplify A Fraction

Example Question #2 : How To Simplify A Fraction

Example Question #3 : How To Simplify A Fraction

Example Question #11 : How To Simplify A Fraction

Example Question #12 : How To Simplify A Fraction

Example Question #13 : How To Simplify A Fraction

Example Question #14 : How To Simplify A Fraction

Example Question #55 : Algebraic Fractions

Example Question #15 : How To Simplify A Fraction

All PSAT Math Resources

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