GMAT Math : Understanding exponents

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Example Questions

Example Question #1 : Exponents

$\frac{6^{3}}{36} + \frac{3^{68}}{3^{67}}=$

Possible Answers:

$\dpi{100} \small 36$

$\dpi{100} \small 18$

$\dpi{100} \small 9$

cannot be determined

$\dpi{100} \small 6$

Correct answer:

$\dpi{100} \small 9$

Explanation:

$\frac{6^{3}}{36} = \frac{6^{3}}{6^{2}} = 6$

$\frac{3^{68}}{3^{67}} = 3^{68-67} = 3$

Putting these together,

$\frac{6^{3}}{36} + \frac{3^{68}}{3^{67}}= 6 + 3 = 9$

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Example Question #1 : Exponents

$\dpi{100} \small 3x^{4}\times x^{2}+x^{2}-x =$

Possible Answers:

$x(3x^{5}-x+1)$

$x(3x^{5}+x-1)$

$3x^{9}$

$3x^{5}+x-1$

$3x^{7}$

Correct answer:

$x(3x^{5}+x-1)$

Explanation:

$\dpi{100} \small 3x^{4}\times x^{2} =3x^{6}$

Then, $\dpi{100} \small 3x^{4}\times x^{2}+x^{2}-x = 3x^{6}+x^{2}-x = x(3x^{5}+x-1)$

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Example Question #2 : Exponents

$4^{\frac{3}{2}} + 27^{\frac{2}{3}} =$

Possible Answers:

$\dpi{100} \small 17$

$\dpi{100} \small 27$

$\dpi{100} \small 8$

$\dpi{100} \small 9$

$\dpi{100} \small 16$

Correct answer:

$\dpi{100} \small 17$

Explanation:

$4^{\frac{3}{2}}=(4^{\frac{1}{2}})^{3} = 2^{3} = 8$

$27^{\frac{2}{3}}=(27^{\frac{1}{3}})^{2} = 3^{2} = 9$

Then putting them together, $4^{\frac{3}{2}} + 27^{\frac{2}{3}} = 8 + 9 = 17$

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Example Question #2 : Exponents

Which of the following expressions is equivalent to this expression?

$\small \frac{x^{-\frac{1}{2}}}{x^{\frac{1}{3}}}$

You may assume that $\small x > 0$ .

Possible Answers:

$\small \frac{\sqrt[6]{x}}{x}$

$\small \small \small \frac{\sqrt[5]{x^{6}}}{x}$

$\small x\sqrt{x}$

$\small \small \frac{\sqrt[6]{x^{5}}}{x}$

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

Correct answer:

$\small \frac{\sqrt[6]{x}}{x}$

Explanation:

$\small \small \small \small \small \small \small \small \small \frac{x^{-\frac{1}{2}}}{x^{\frac{1}{3}}} = x^{-\frac{1}{2} -{\frac{1}{3}} } = x^{-\frac{2}{6} -{\frac{3}{6}} }= x^{-\frac{5}{6} }= \frac{1}{ x^{\frac{5}{6}}}$

$\small \small \small = \frac{1}{\sqrt[6]{x^{5}}}= \frac{\sqrt[6]{x}}{\sqrt[6]{x^{5}}\cdot \sqrt[6]{x}}= \frac{\sqrt[6]{x}}{x}$

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Example Question #3 : Exponents

Simplify the following expression without a calculator:

$(\frac{9x^\frac{-5}{2}y^\frac{7}{4}}{x^\frac{3}{2}y^\frac{-1}{4}})^\frac{3}{2}$

Possible Answers:

$27x^\frac{-3}{2}y^\frac{9}{4}$

$\frac{27y^3}{x^6}$

$9x^\frac{-3}{2}y^\frac{9}{4}$

$\frac{9y^3}{x^6}$

$\frac{27y^3}{x^3}$

Correct answer:

$\frac{27y^3}{x^6}$

Explanation:

The easiest way to simplify is to work from the inside out. We should first get rid of the negatives in the exponents. Remember that variables with negative exponents are equal to the inverse of the expression with the opposite sign. For example, $a^\frac{-5}{2} = \frac{1}{a^\frac{5}{2}}$ So using this, we simplify: $(\frac{9x^\frac{-5}{2}y^\frac{7}{4}}{x^\frac{3}{2}y^\frac{-1}{4}})^\frac{3}{2} = (\frac{9(y^\frac{7}{4})(y^\frac{1}{4})}{(x^\frac{5}{2})(x^\frac{3}{2})})^\frac{3}{2}$

Now when we multiply variables with exponents, to combine them, we add the exponents together. For example, $(a^2)(a^2)=a^{2+2} = a^4$

Doing this to our expression we get it simplified to $(\frac{9y^2}{x^4})^\frac{3}{2}$ .

The next step is taking the inside expression and exponentiating it. When taking an exponent of a variable with an exponent, we actually multiply the exponents. For example, $(a^2)^4 = a^{(2*4)}= a^8$ . The other rule we must know that is an exponent of one half is the same as taking the square root. So for the $9^{3/2}=(\sqrt9)^3$ So using these rules, $(\frac{9y^2}{x^4})^\frac{3}{2} = \frac{9^{3/2}(y^2)^{3/2}}{(x^4)^{3/2}} = \frac{(\sqrt9)^3y^{(2*3/2)}}{x^{(4*3/2)}} = \frac{3^3y^3}{x^6}= \frac{27y^3}{x^6}$

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Example Question #3 : Exponents

Rewrite as a single logarithmic expression:

$3 + \log_{3} x - 2\log_{3} y$

Possible Answers:

$\log_{3} \frac{3x}{2y}$

$\log_{3} (3 + x -2y )$

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

$\log_{3} \frac{27x}{2y}$

$\log_{3} \frac{27x}{y^{2}}$

Correct answer:

$\log_{3} \frac{27x}{y^{2}}$

Explanation:

First, write each expression as a base 3 logarithm:

$3 = \log_{3}27$ since $3^{3} = 27$

$2 \log_{3}y = \log_{3} y^{2}$

Rewrite the expression accordingly, and apply the logarithm sum and difference rules:

$3 + \log_{3} x - 2\log_{3} y$

$=\log_{3}27 + \log_{3} x - \log_{3} y^{2}$

$=\log_{3}\left (27 \cdot x \right )- \log_{3} y^{2}$

$=\log_{3}\left (27 \cdot x \right \div y^{2} ) = \log_{3}\left (\frac{27 x }{y^{2}}\right )$

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Example Question #1 : Exponents

If 3^p+3^p+3^p=3^q , what is in terms of ?

Possible Answers:

q-1

q+1

$\frac{q}{6}$

$\frac{q}{3}$

Correct answer:

q-1

Explanation:

We have $3^p+3^p+3^p=3*3^p=3^{p+1}=3^q$ .

So p+1=q , and p=q-1 .

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Example Question #2 : Exponents

What are the last two digits, in order, of $6 ^{789}$ ?

Possible Answers:

Correct answer:

Explanation:

Inspect the first few powers of 6; a pattern emerges.

$6^{1} = 6$

$6^{2} = 36$

$6^{3} = 216$

$6^{4} = 1,296$

$6^{5} = 7,776$

$6^{6} = 46,656$

$6^{7} = 279,936$

$6^{8} = 1,679,616$

$6^ { 9 } =10,077,696$

$6^ { 10} =60,466,176$

As you can see, the last two digits repeat in a cycle of 5.

789 divided by 5 yields a remainder of 4; the pattern that becomes apparent in the above list is that if the exponent divided by 5 yields a remainder of 4, then the power ends in the diigts 96.

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Example Question #31 : Algebra

Which of the following expressions is equal to the expression

$\left ( x^{3}\right ) ^{4} \cdot \left ( x^{3}\right ) ^{5}$ ?

Possible Answers:

$x^{12}$

$x^{27}$

$x^{180}$

$x^{60}$

$x^{15}$

Correct answer:

$x^{27}$

Explanation:

Use the properties of exponents as follows:

$\left ( x^{3}\right ) ^{4} \cdot \left ( x^{3}\right ) ^{5}$

$=x^{3\; \cdot \; 4} \cdot x^{3\; \cdot \; 5}$

$=x^{12} \cdot x^{15}$

$=x^{12+15}$

$=x^{27}$

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Example Question #2 : Exponents

Simplify:

$\left [ \left ( x ^{5} \right )^{4} \right ]^{3}$

Possible Answers:

12x

$x ^{12}$

60x

$60x ^{12}$

$x ^{60}$

Correct answer:

$x ^{60}$

Explanation:

Apply the power of a power principle twice by multiplying exponents:

$\left [ \left ( x ^{5} \right )^{4} \right ]^{3} = \left ( x ^{5 \cdot 4} \right ) ^{3} = \left ( x ^{20} \right ) ^{3} = x ^{20 \cdot 3} = x ^{60 }$

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GMAT Math : Understanding exponents

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Exponents

Example Question #1 : Exponents

Example Question #2 : Exponents

Example Question #2 : Exponents

Example Question #3 : Exponents

Example Question #3 : Exponents

Example Question #1 : Exponents

Example Question #2 : Exponents

Example Question #31 : Algebra

Example Question #2 : Exponents

All GMAT Math Resources

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