GMAT Math : Algebra

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Example Questions

Example Question #23 : Simplifying Algebraic Expressions

Simplify:

$\left ( x - 1\right ) + \left ( x-1\right )^{2} + \left ( x-1\right )^{3}$

Possible Answers:

$x^{3}-2x^{2} -4x+2$

$x^{3}+4x^{2} -4x+2$

$x^{3}-2 x^{2} + 2x- 1$

$x^{3}-2 x^{2} + 2x+2$

$x^{3}+4x^{2} -4x- 1$

Correct answer:

$x^{3}-2 x^{2} + 2x- 1$

Explanation:

$\left (x-1 \right )^{2}$

$= x^{2}- 2 \cdot 1 \cdot x + 1^{2}$

$= x^{2}- 2 x + 1$

$\left (x-1 \right )^{3}$

$= x^{3} - 3 \cdot 1 \cdot x^{2} + 3 \cdot 1^{2} \cdot x - 1^{3}$

$= x^{3} -3 x^{2} + 3 x - 1$

$\left ( x - 1\right ) + \left ( x-1\right )^{2} + \left ( x-1\right )^{3}$

$= \left ( x - 1\right ) + \left (x^{2}- 2 x + 1\right ) + \left (x^{3} -3 x^{2} + 3 x - 1\right )$

$= x^{3}+x^{2} -3 x^{2} + x -2x+3x- 1+1-1$

$= x^{3}-2 x^{2} + 2x- 1$

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Example Question #24 : Simplifying Algebraic Expressions

Simplify: $(x-7)^{2} - (x+14)^{2}$

Possible Answers:

14x-147

-42x -147

-21x -147

14x+245

-42x +245

Correct answer:

-42x -147

Explanation:

Use the perfect square trinomial pattern to expand, then collect like terms:

$(x-7)^{2} - (x+14)^{2}$

$=(x^{2} - 2\cdot 7 \cdot x +7^{2} ) - (x^{2} + 2\cdot 14 \cdot x +14^{2} )$

$=(x^{2} -14 x +49) - (x^{2} + 28x +196 )$

$=x^{2} -14 x +49 - x^{2} - 28x -196$

$=x^{2}- x^{2} -14 x- 28x +49 -196$

= -42x -147

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Example Question #25 : Simplifying Algebraic Expressions

Simplify: $(x+3)+(x+3)^{2}+ (x+3)^{3}$

Possible Answers:

$x^{3}+ 10x^{2} +34 x + 39$

$x^{3}+ 9 x^{2} + 27 x + 38$

$x^{3}+ 9x^{2} +34 x + 39$

$x^{3}+ 9 x^{2} + 27 x + 39$

$x^{3}+10 x^{2} + 27 x + 38$

Correct answer:

$x^{3}+ 10x^{2} +34 x + 39$

Explanation:

Expand $(x+3)^{2}$ and $(x+3)^{3}$ :

$(x+3)^{2}$

$= x^{2}+ 2 \cdot x \cdot 3 + 3^{2}$

$= x^{2}+ 6x + 9$

$(x+3)^{3}$

$= x^{3}+ 3 \cdot x^{2} \cdot 3 + 3 \cdot x\cdot 3 ^{2} + 3 ^{3}$

$= x^{3}+ 9 x^{2} + 27 x + 27$

Add:

$x^{3}+ 9 x^{2} + 27 x + 27$

$x^{2}+\; 6x +\; \; \; 9$

$\underline{ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\; \; \; +3}$

$x^{3}+ 10x^{2} +34 x + 39$

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Example Question #1337 : Problem Solving Questions

Which of the following is equal to $\sqrt{6x-6} \cdot \sqrt{15x-15} \cdot \sqrt{35x-35}$ ?

Possible Answers:

$15x\sqrt{14x-14}-15\sqrt{14x-14}$

$14x\sqrt{15x-15}-14\sqrt{15x-15}$

$210 x\sqrt{x-1}-210 \sqrt{x-1}$

210x-210

$210x^{2}-420x+210$

Correct answer:

$15x\sqrt{14x-14}-15\sqrt{14x-14}$

Explanation:

Factor all radicands, multiply, then simplify:

$\sqrt{6x-6} \cdot \sqrt{15x-15} \cdot \sqrt{35x-35}$

$= \sqrt{2 \cdot 3\cdot (x-1)} \cdot \sqrt{3 \cdot 5 \cdot (x-1)} \cdot \sqrt{5\cdot 7\cdot (x-1)}$

$= \sqrt{2 \cdot 3\cdot3 \cdot 5 \cdot 5\cdot 7\cdot(x-1)\cdot (x-1)\cdot (x-1)}$

$=\sqrt{3\cdot3} \cdot \sqrt{5 \cdot 5} \cdot \sqrt{(x-1)\cdot (x-1)} \cdot \sqrt{2 \cdot 7 \cdot (x-1)}$

$=3 \cdot5 \cdot (x-1) \cdot \sqrt{14x-14}$

$= (15x-15) \cdot \sqrt{14x-14}$

$= 15x\sqrt{14x-14}-15\sqrt{14x-14}$

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Example Question #21 : Simplifying Algebraic Expressions

Simplify the expression:

(x-1)(x+1)+(x-2)(x+2)+(x-3)(x+3)

Possible Answers:

$3x^{2}-11$

$3x^{2}-6x -1 4$

$3x^{2}-12x -1 4$

$3x^{2}$

$3x^{2}-1 4$

Correct answer:

$3x^{2}-1 4$

Explanation:

Each of the three products added is a product of the sum and the difference of the same two expressions, so each can be simplified using that pattern:

$(x-1)(x+1)= x^{2}- 1^{2} = x^{2}- 1$

$(x-2)(x+2)= x^{2}- 2^{2} = x^{2}-4$

$(x-3)(x+3)= x^{2}- 3^{2} = x^{2}-9$

Add:

(x-1)(x+1)+(x-2)(x+2)+(x-3)(x+3)

$=(x^{2}-1) +(x^{2}-4) +(x^{2}-9)$

$=x^{2}+x^{2}+x^{2}-1 -4 -9$

$=3x^{2}-1 4$

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Example Question #21 : Simplifying Algebraic Expressions

Simplify: $\sqrt[3]{ \sqrt[4]{x ^{20}} }$

You may assume is positive.

Possible Answers:

$x^{8}$

$x^{2} \sqrt[7]{x^{6}}$

$x^{13}$

None of the other choices gives the correct answer.

$x\sqrt[3]{x^{2}}$

Correct answer:

$x\sqrt[3]{x^{2}}$

Explanation:

$\sqrt[m]{x^{n}}$ is equal to $x^{\frac{m}{n}}$ , so $\sqrt[4]{x ^{20}}$ can be rewritten as $x^{\frac{20}{4}}$ or $x^{5}$ .

The original expression can be rewritten as $\sqrt[3]{ x ^{5}}$ , which itself can be rewritten by noting that 5 divided by 3 yields quotient 1 and remainder 2. Therefore, the simplified form is $x^{1} \sqrt[3]{x^{2}}$ , or $x \sqrt[3]{x^{2}}$ .

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Example Question #1340 : Problem Solving Questions

Simplify the expression:

$\sqrt[5]{x^{-8}y^{7}}$

Possible Answers:

$\frac{y \sqrt[5]{ x^{2} y^{2}} }{ x^{2}}$

$\frac{y \sqrt[5]{ x y^{2}} }{ x^{2}}$

$\frac{y \sqrt[5]{ x^{2} y^{2}} }{ x }$

$\frac{y \sqrt[5]{ x^{3} y^{2}} }{ x^{2}}$

$\frac{y \sqrt[5]{ x y^{2}} }{ x }$

Correct answer:

$\frac{y \sqrt[5]{ x^{2} y^{2}} }{ x^{2}}$

Explanation:

First, rewrite the radicand with positive exponents, then take the fifth root of the numerator and the denominator:

$\sqrt[5]{x^{-8}y^{7}} = \sqrt[5]{\frac{y^{7}}{x^{8}}} = \frac{ \sqrt[5]{ y^{7}} }{ \sqrt[5]{ x^{8}}}$

To simplify each expression, divide the exponent by the index 5, and note its quotient and remainder. 7 divided by 5 is 1 with remainder 2; 8 divided by 5 is 1 with remainder 3. Subsequently,

$\frac{ \sqrt[5]{ y^{7}} }{ \sqrt[5]{ x^{8}}} =\frac{y \sqrt[5]{ y^{2}} }{x \sqrt[5]{ x^{3}}}$

The denominator can be rationalized by multiplying both halves by $\sqrt[5]{ x^{2}}$ :

$\frac{y \sqrt[5]{ y^{2}}\cdot \sqrt[5]{ x^{2}} }{x \sqrt[5]{ x^{3}}\cdot \sqrt[5]{ x^{2}}}$

$=\frac{y \sqrt[5]{ x^{2} y^{2}} }{x \cdot \sqrt[5]{ x^{5}}}$

$=\frac{y \sqrt[5]{ x^{2} y^{2}} }{x \cdot x}$

$=\frac{y \sqrt[5]{ x^{2} y^{2}} }{ x^{2}}$

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Example Question #26 : Simplifying Algebraic Expressions

Simplify:

$(x+5)(x^{2}-5x+25) - (x-5)(x^{2}+5x+25)$

Possible Answers:

The correct answer is not among the other choices.

$25x^{2}+250$

$2x^{3} +10x$

$2x^{3} +50x$

$5x^{2}+250$

Correct answer:

The correct answer is not among the other choices.

Explanation:

The problem is easier if you recognize the two products $(x+5)(x^{2}-5x+25)$ and $(x-5)(x^{2}+5x+25)$ as the factorizations of the sum and difference of two cubes, respectively:

$(x+5)(x^{2}-5x+25) = (x+5)(x^{2}-5 \cdot x +5^{2}) = x^{3}+5^{3}= x^{3}+125$

$(x-5)(x^{2}+5x+25) = (x-5)(x^{2}+5 \cdot x +5^{2}) = x^{3}-5^{3}= x^{3}-125$

Therefore,

$(x+5)(x^{2}-5x+25) - (x-5)(x^{2}+5x+25)$

$= (x^{3} +125) - (x^{3} -125)$

$= x^{3} +125 - x^{3} +125$

= 250

This is not one of the responses.

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Example Question #31 : Simplifying Algebraic Expressions

Simplify: $\left (\sqrt[6]{x^{24}} \right )^{-4}$ . Assume is nonnegative.

Possible Answers:

$x^{26}$

$\frac{1}{x^{80}}$

$\frac{1}{x^{16}}$

$\frac{1}{x^{ 6}}$

Correct answer:

$\frac{1}{x^{16}}$

Explanation:

$\sqrt[m]{x^{n}}$ is equal to $x^{\frac{m}{n}}$ , $\left (x^{m} \right )^{n} = x^{mn}$ , and $x^{-n}=\frac{1}{x^{n}}$ ; combine these three properties as follows:

$\left (\sqrt[6]{x^{24}} \right )^{-4} = \left ( x^{\frac{24}{6}} \right )^{-4} = \left ( x^{4} \right )^{-4} = x^{4 \cdot (-4)} = x^{-16} = \frac{1}{x^{16}}$

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Example Question #32 : Simplifying Algebraic Expressions

Simplify:

$(x+7)(x^{2}-7x+49) + (x-7)(x^{2}+7x+49)$

Possible Answers:

$2x^{3} + 14x ^{2}+ 98x+ 686$

$2x^{3}$

$2x^{3} + 98x$

$2x^{3} - 14x ^{2}+ 98x- 686$

$2x^{3} + 686$

Correct answer:

$2x^{3}$

Explanation:

The problem is easier if you recognize the two products $(x+7)(x^{2}-7x+49)$ and $(x-7)(x^{2}+7x+49)$ as the factorizations of the sum and difference of two cubes, respectively:

$(x+7)(x^{2}-7x+49)= (x+7)(x^{2}-7 \cdot x +7^{2}) = x^{3}+7^{3}= x^{3}+343$

$(x-7)(x^{2}+7x+49)= (x-7)(x^{2}+7 \cdot x +7^{2}) = x^{3}-7^{3}= x^{3}-343$

Therefore,

$(x+7)(x^{2}-7x+49) + (x-7)(x^{2}+7x+49)$

$=\left ( x^{3}+343 \right )+ \left ( x^{3}-343 \right )$

$= x^{3}+x^{3}+343 -343$

$=2x^{3}$

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #23 : Simplifying Algebraic Expressions

Example Question #24 : Simplifying Algebraic Expressions

Example Question #25 : Simplifying Algebraic Expressions

Example Question #1337 : Problem Solving Questions

Example Question #21 : Simplifying Algebraic Expressions

Example Question #21 : Simplifying Algebraic Expressions

Example Question #1340 : Problem Solving Questions

Example Question #26 : Simplifying Algebraic Expressions

Example Question #31 : Simplifying Algebraic Expressions

Example Question #32 : Simplifying Algebraic Expressions

All GMAT Math Resources

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