GMAT Math : Algebra

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Example Questions

Example Question #51 : Algebra

$M = \log A$

$N = \log B$

$P = \log C$

Which of the following is equal to $\log \frac{100 A^{2} B}{C ^{3}}$ ?

Possible Answers:

100 + 2 M + N - 3P

$100 + M^{2} + N - P^{3}$

$\frac{ 4 M N }{ 3P}$

2 + 2 M + N - 3P

$2 + M^{2} + N - P^{3}$

Correct answer:

2 + 2 M + N - 3P

Explanation:

We'll need to remember a few logarithmic properties to answer this question:

$log(A\times B)=log(A)+log(B)$

$log(\frac{A}{B})=log(A)-log(B)$

$log(A^x)=x\cdot log(A)$

Now we can use these same rules to rewrite the log in question:

$\log \frac{100 A^{2} B}{C ^{3}}$

$= \log \left ( 100 A^{2} B \right )- \log \left ( C ^{3} \right )$

$= \log 100 + \log \left ( A^{2} \right ) + \log B - \log \left ( C ^{3} \right )$

$= \log 100 + 2 \log A + \log B - 3 \log C$

=2 + 2 M + N - 3P

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Example Question #51 : Algebra

Simplify the following expression:

$\frac{(x^{2+3})^{4}}{x^{12}}$ .

Possible Answers:

$x^{8}$

$x^{-3}$

$x^{32}$

$x^{\frac{5}{3}}$

$x^{2}$

Correct answer:

$x^{8}$

Explanation:

We start by simplifying the expression on the top. Let's add the exponents inside the parentheses and then multiply by the exponent outside of the parentheses.

Then we substract the denominator exponent form the numerator exponent:

$\frac{(x^{2+3})^{4}}{x^{12}}=\frac{(x^{5})^{4}}{x^{12}}=\frac{x^{5*4}}{x^{12}}=\frac{x^{20}}{x^{12}}=x^{20-12}=x^{8}$

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Example Question #52 : Algebra

Simplify the following:

$a^{6}b^{-4}c^{-2}\cdot \frac{a^7b^{11}c^{13}}{a^{13}b^{7}c^{11}}}$ .

Possible Answers:

$a^{26}b^{14}c^{26}$

abc

$a^{14}b^{8}c^{4}$

$a^{-13}b^{13}c^{13}$

Correct answer:

Explanation:

First, we need to add the exponents of the elements with the same base that are multiplied, and subtract the exponents of same-base elements that are divided:

$\frac{a^7b^{11}c^{13}}{a^{13}b^{7}c^{11}}}=a^{-6}b^{4}c^{2}$

Then $a^{6}b^{-4}c^{-2}\cdot a^{-6}b^{4}c^{-2}=a^{0}b^{0}c^{0} or (abc)^{0}}$

Any value raised to the power of 0 equals 1 so the final result is 1.

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Example Question #31 : Understanding Exponents

0<x<1 . Order from least to greatest: $x^{-\frac{1}{2}}, x^{-\frac{1}{3}}, x^{-\frac{1}{4}}, x^{-\frac{1}{5}}$ .

Possible Answers:

$x^{-\frac{1}{5}}, x^{-\frac{1}{3}}, x^{-\frac{1}{4}}, x^{-\frac{1}{2}}$

$x^{-\frac{1}{5}}, x^{-\frac{1}{4}}, x^{-\frac{1}{3}}, x^{-\frac{1}{2}}$

This is impossible, since at least one expression is undefined.

$x^{-\frac{1}{3}}, x^{-\frac{1}{5}}, x^{-\frac{1}{2}}, x^{-\frac{1}{4}}$

$x^{-\frac{1}{2}}, x^{-\frac{1}{3}}, x^{-\frac{1}{4}}, x^{-\frac{1}{5}}$

Correct answer:

$x^{-\frac{1}{5}}, x^{-\frac{1}{4}}, x^{-\frac{1}{3}}, x^{-\frac{1}{2}}$

Explanation:

It might be easiest to order $x^{\frac{1}{2}}, x^{\frac{1}{3}}, x^{\frac{1}{4}}, x^{\frac{1}{5}}$ first and work from there.

Since is a positive number less than 1, if M > N > 0 , then $x^{M }< x^{N}$ . Therefore, those four expressions, from least to greatest, are

$x^{\frac{1}{2}}, x^{\frac{1}{3}}, x^{\frac{1}{4}}, x^{\frac{1}{5}}$ .

If $x^{M }< x^{N}$ , then $\frac{1}{x^{M }}> \frac{1}{x^{N}}$ - that is, $x^{-M}> x^{-N}$ . So changing the signs of the exponents reverses the order. As a result, the orginal four expressions, in ascending order, are

$x^{-\frac{1}{5}}, x^{-\frac{1}{4}}, x^{-\frac{1}{3}}, x^{-\frac{1}{2}}$

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Example Question #32 : Exponents

-1< x< 0 . Order from least to greatest: $x^{-1}, x^{-2}, x^{-3}, x^{-4}$ .

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

If is negative, then:

Even powers $x^{2}$ and $x^{4}$ are positive, with $x^{2} = |x| ^{2}$ and $x^{4} = |x| ^{4}$ .

Since $x \in \left (-1, 0 \right )$ , |x| < 1 ,

It follows that $|x|^{4}<| x|^{2}$ , and $x^{4}<x^{2}$ .

If $x^{M }< x^{N}$ , then $\frac{1}{x^{M }}> \frac{1}{x^{N}}$ - that is, $x^{-M}> x^{-N}$ . So changing the signs of the exponents reverses the order. As a result,

$x^{-2} < x^{-4}$ .

Odd powers and $x^{3}$ are negative, with $x =- |x| ^{1}$ and $x^{3} = - |x| ^{3}$ .

$|x| ^{3} < |x|^{1}$ , so $-|x| ^{3} > -|x|^{1}$ , and $x^{1}< x^{3}$ .

As before, changing their exponents to their opposites reverses the order:

$x^{-3}< x^{-1}$

Setting the negative numbers less than the positive numbers:

$x^{-3}< x^{-1}<x^{-2}< x^{-4}$ .

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Example Question #32 : Understanding Exponents

0<x<1 . Order from least to greatest: $x^{\frac{1}{2}}, x^{\frac{1}{3}}, x^{\frac{1}{4}}, x^{\frac{1}{5}}$ .

Possible Answers:

$x^{\frac{1}{5}},x^{\frac{1}{3}}, x^{\frac{1}{2}}, x^{\frac{1}{4}}$

$x^{\frac{1}{3}}, x^{\frac{1}{5}},x^{\frac{1}{4}}, x^{\frac{1}{2}}$

$x^{\frac{1}{2}}, x^{\frac{1}{3}}, x^{\frac{1}{4}}, x^{\frac{1}{5}}$

$x^{\frac{1}{5}}, x^{\frac{1}{4}}, x^{\frac{1}{3}}, x^{\frac{1}{2}}$

$x^{\frac{1}{3}}, x^{\frac{1}{5}}, x^{\frac{1}{2}}, x^{\frac{1}{4}}$

Correct answer:

$x^{\frac{1}{2}}, x^{\frac{1}{3}}, x^{\frac{1}{4}}, x^{\frac{1}{5}}$

Explanation:

Since is a positive number less than 1, if M > N > 0 , then $x^{M }< x^{N}$ . Therefore,

$x^{\frac{1}{2}}< x^{\frac{1}{3}}< x^{\frac{1}{4}}<x^{\frac{1}{5}}$ .

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Example Question #33 : Understanding Exponents

Simplify.

$\frac{x^{5}y^{10}z^{-4}}{x^{3}y^{7}z}$

Possible Answers:

x^2y^3z^3

$\frac{x^8y^{17}}{z^5}$

$\frac{x^2y^3}{z^3}$

$\frac{x^2y^3}{z^5}$

Correct answer:

$\frac{x^2y^3}{z^5}$

Explanation:

In order to solve this problem we have to keep in mind three properties of exponents:

$\frac{x^n}{x^m}=x^{n-m}$
$x^{-m}=\frac{1}{x^{m}}$
$x^{m}x^{n}=x^{m+n}$

The first thing we can do is take care of the negative exponent:

$\frac{x^{5}y^{10}z^{-4}}{x^{3}y^{7}z} = \frac{x^{5}y^{10}}{x^{3}y^{7}z^1 z^4}$

I included a as the exponent for in order to make the calculation easier to see.

We can now simplify by looking into each variable individually:

$\frac{x^5}{x^3}=x^2$

$\frac{y^{10}}{y^7}=y^3$

$\frac{1}{z^1 z^4}=\frac{1}{z^{1+4}}=\frac{1}{z^5}$

The last step is to put these answers together:

$\frac{x^2y^3}{z^5}$

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Example Question #34 : Understanding Exponents

Simplify.

$\frac{3x^9y^{-2}z^7}{18x^8y^{-3}z}$

Possible Answers:

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

$\frac{xz^6}{6y^5}$

6xyz^6

$\frac{3xz^6}{18y}$

Correct answer:

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

Explanation:

In order to solve this problem we have to keep in mind three properties of exponents:

$\frac{x^n}{x^m}=x^{n-m}$
$x^{-m}=\frac{1}{x^m}$
$x^mx^n=x^{m+n}$

The first thing we can do is take care of the negative exponents:

$\frac{3x^9y^{-2}z^7}{18x^8y^{-3}z} = \frac{3x^9y^{3}z^7}{18x^8y^{2}z^1}$

We can now simplify by working on each variable individually:

$\frac{x^9}{x^8}=x^{9-8}=x^1$

$\frac{y^3}{y^2}=y^{3-2}=y^1$

$\frac{z^7}{z^1}=z^{7-1}=z^6$

Don't forget to simplify the coefficients as well: $\frac{3}{18}=\frac{1}{6}$

We can now arrive to our answer by combining these:

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

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Example Question #53 : Algebra

Which of the following numbers is in scientific notation?

Possible Answers:

$6.7 \times 10^{-9}$

$0.067 \times 10^{-4}$

$0.67 \times 10^{-8}$

$670 \times 10^{-5}$

$67 \times 10^{-8}$

Correct answer:

$6.7 \times 10^{-9}$

Explanation:

The number in front must have an absolute value greater than or equal to 1, and less than 10. Of these choices, only $6.7 \times 10^{-9}$ qualifies.

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Example Question #36 : Understanding Exponents

0<x<1 . Order from least to greatest: $x, x^{2}, x^{3}, x^{4}$ .

Possible Answers:

$x^{2}, x^{4} , x^{3},x$

$x^{4} , x^{3}, x^{2}, x$

$x^{4} , x^{2}, x^{3},x$

$x^{2}, x^{4} , x,x^{3}$

$x, x^{2}, x^{3}, x^{4}$

Correct answer:

$x^{4} , x^{3}, x^{2}, x$

Explanation:

If 0 < x < 1 , then if M >N , $x^{M} < x^{N}$ . Therefore,

$x^{4} < x^{3}< x^{2} < x$ .

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #51 : Algebra

Example Question #51 : Algebra

Example Question #52 : Algebra

Example Question #31 : Understanding Exponents

Example Question #32 : Exponents

Example Question #32 : Understanding Exponents

Example Question #33 : Understanding Exponents

Example Question #34 : Understanding Exponents

Example Question #53 : Algebra

Example Question #36 : Understanding Exponents

All GMAT Math Resources

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