GMAT Math : Algebra

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Example Questions

Example Question #57 : Understanding Exponents

Fill in the circle with a number so that this polynomial is prime:

$x^{2}+ \bigcirc x- 60$

Possible Answers:

Correct answer:

Explanation:

If $x^{2}+ B x- 60$ is not prime, then, as a quadratic trinomial of the form $x^{2}+ B x- C$ it is factorable as

(x+m)(x-n)

where mn = 60 and m - n = B .

Therefore, we are looking for a whole number that is not the difference of two factors of 60. The integers that are such a difference are

60 -1= 59

30-2=28

20-3=17

15-4=11

12-5 = 7

10 -6 = 4

Of the choices, only 23 is not on the list, so it is the correct choice.

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Example Question #58 : Understanding Exponents

Examine these two polynomials, each of which is missing a number:

$x^{2}-8 x + \square$

$x^{3}+ \bigcirc$

Write a whole number inside each shape so that the first polynomial is a factor of the second.

Possible Answers:

Write 64 in the square and 4.096 in the circle

Write 8 in the square and 64 in the circle

Write 64 in the square and 512 in the circle

Write 64 in the square and 64 in the circle

Write 8 in the square and 512 in the circle

Correct answer:

Write 64 in the square and 512 in the circle

Explanation:

The factoring pattern to look for in the second polynomial is the sum of cubes, so the number in the circle must be a perfect cube of a whole number . We can write this polynomial as

$x^{3}+ B^{3}$

which can be factored as

$(x+B) (x^{2}-Bx+B^{2})$

By the condition of the problem, B = 8 , so the number that replaces the square is $8^{2} = 64$ , and the number that replaces the circle is $8^{3} = 512$ .

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Example Question #81 : Algebra

Simplify.

$\frac{3x^{-4}y^2zx^2}{21yz^{-3}}$

Possible Answers:

$\frac{yz^4}{7x^2}$

${7x^2yz^4}$

$\frac{x^2y}{7z^2}$

$\frac{y}{7x^2z^4}$

Correct answer:

$\frac{yz^4}{7x^2}$

Explanation:

There are different ways to approach this problem. We just need to remember three things:

$\frac{x^a}{x^b}=x^{a-b}$
$x^{-a}=\frac{1}{x^a}$
$x^ax^b=x^{a+b}$

Keeping those in mind, we can simplify the numerator and coefficients:

$\frac{3x^{-4}y^2zx^2}{21yz^{-3}}=\frac{1x^{-4+2}y^2z}{7yz^{-3}}=\frac{1x^{-2}y^2z}{7yz^{-3}}$

I'm going to move the negative exponents (number 2 in the list above) in order to make them positive:

$\frac{1\mathbf{x^{-2}}y^2z}{7y\mathbf{z^{-3}}}=\frac{z^3y^2z}{7x^2y}$

We can now simplify the numerator again with the exponents (number 3) and the exponents (number 1):

$\frac{z^3y^2z}{7x^2y}=\frac{yz^4}{7x^2}$

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Example Question #61 : Understanding Exponents

Simplify:

$\frac{(4x^{7}y^{-4}z^{0})^{3}}{(2x^{-3}y^{10}z^1)^4}$

Possible Answers:

$\frac{4x^{33}}{y^{52}z^{4}}}$

Unable to simplify

$\frac{64x^{21}y^{-12}}{16x^{-12}y^{40}z^4}$

$\frac{4x^9}{y^{13}z^2}$

$\frac{64x^{10}y^{-1}z^3}{16xy^{14}z^5}$

Correct answer:

$\frac{4x^{33}}{y^{52}z^{4}}}$

Explanation:

The first thing we must do is distribute the exponents outside of the parentheses across each expression (remembering of course that exponents set to another exponent are multiplied).

$\frac{(4x^{7}y^{-4}z^{0})^{3}}{(2x^{-3}y^{10}z^1)^4} = \frac{4^3(x^7)^3(y^{-4})^3(z^0)^3}{2^4(x^{-3})^4(y^{10})^4(z^1)^4} =$ $\frac{64x^{21}y^{-12}}{16x^{-12}y^{40}z^4}$

The last step is to follow the rules of exponent addition/subtraction:

$a^b \cdot a^c = a^{b+c}$ and $a^b / a^c = a^{b-c}$

Therefore:

$\frac{64x^{21}y^{-12}}{16x^{-12}y^{40}z^4} =$ $\frac{4x^{33}}{y^{52}z^{4}}}$

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Example Question #62 : Exponents

Which of the following is true if $5^{x}= 3,899$ ?

Possible Answers:

8 < x < 9

7 < x < 8

The equation has no solution.

5 < x < 6

6< x < 7

Correct answer:

5 < x < 6

Explanation:

To answer this question, note that

$5^{5}= 3,125$

and

$5^{6}= 15,625$ .

Therefore, since

3,125< 3,899<15,625

it follows that

$5^{5} < 5^{x}< 5^{6}$

and

5 < x < 6 .

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Example Question #62 : Understanding Exponents

Solve for :

$4^{x}= 512$

Possible Answers:

$x= \frac{9}{2}$

x = 4

The equation has no solution.

$x= \frac{1}{4}$

$x= \frac{2}{9}$

Correct answer:

$x= \frac{9}{2}$

Explanation:

Rewrite both sides as powers of 2 using the exponent rules as follows:

$4^{x}= 512$

$(2^{2} )^{x}= 512$

$2 ^{2\cdot x} = 2^{9}$

Since powers of the same base are equal, set the exponents equal to each other:

2x = 9

$x= \frac{9}{2}$

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Example Question #63 : Understanding Exponents

Solve for :

$9^{t} = 3^{t - 4}$

Possible Answers:

t = 4

t = 2

The equation has no solution.

t = -4

t= -2

Correct answer:

t = -4

Explanation:

Rewrite both sides as powers of 3 using the exponent rules as follows:

$9^{t} = 3^{t - 4}$

$(3^{2})^{t} = 3^{t - 4}$

$3^{2\cdot t}= 3^{t - 4}$

Since powers of the same base are equal, set the exponents equal to each other:

2t = t-4

2t - t= t-4 - t

t = -4

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Example Question #64 : Understanding Exponents

Solve for :

$2^{x+1} \cdot 4 ^{x} = 8^{3x}$

Possible Answers:

x = 1

x= 6

$x= \frac{3}{2}$

The equation has no solution.

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

Correct answer:

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

Explanation:

Rewrite all expressions as powers of 2, and use the exponent rules as follows:

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

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

$2^{x+1} \cdot 2^{2x} = 2^{9x }$

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

$2^{3x+1 } = 2^{9x }$

Since powers of the same base are equal, set the exponents equal to each other:

3x+1=9x

3x+1 - 3x=9x - 3x

1 = 6x

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

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Example Question #1172 : Gmat Quantitative Reasoning

Solve for :

$2 ^{x} + 2 ^{x-1} = 24$

Possible Answers:

$x = \frac{9}{2}$

x= 4

The equation has no solution.

x= 3

$x= \frac{7}{2}$

Correct answer:

x= 4

Explanation:

Rewrite the first expression to get:

$2 ^{x} + 2 ^{x-1} = 24$

$2 \cdot 2 ^{x-1} + 2 ^{x-1} = 24$

$2 \cdot 2 ^{x-1} + 1 \cdot 2 ^{x-1} = 24$

$3 \cdot 2 ^{x-1} = 24$

$\frac{3 \cdot 2 ^{x-1} }{3}= \frac{24}{3}$

$2 ^{x-1} = 8$

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

x-1 = 3

x=4

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Example Question #65 : Understanding Exponents

Solve for :

$2^{3t} = 8^{t - 4}$

Possible Answers:

t = 0

t = 3

t = -4

t = 1

The equation has no solution.

Correct answer:

The equation has no solution.

Explanation:

Rewrite both sides as powers of 2 using the exponent rules as follows:

$2^{3t} = 8^{t - 4}$

$2^{3t} = (2^{3}) ^{t - 4}$

$2^{3t} = 2^{3\left ( t - 4 \right )}$

Since powers of the same base are equal, set the exponents equal to each other:

3t = 3t-12

3t - 3t= 3t-12 - 3t

0 = -12

This is identically false, so the equation has no solution.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #57 : Understanding Exponents

Example Question #58 : Understanding Exponents

Example Question #81 : Algebra

Example Question #61 : Understanding Exponents

Example Question #62 : Exponents

Example Question #62 : Understanding Exponents

Example Question #63 : Understanding Exponents

Example Question #64 : Understanding Exponents

Example Question #1172 : Gmat Quantitative Reasoning

Example Question #65 : Understanding Exponents

All GMAT Math Resources

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