Understanding exponents - GMAT Quantitative

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Question

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

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Answer

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

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

Putting these together,

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

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Question

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

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Answer

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

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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Question

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

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Answer

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

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

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

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Question

If , what is in terms of ?

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Answer

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

So , and .

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Question

Which of the following expressions is equivalent to this expression?

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

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

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Answer

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

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

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Question

Simplify the following expression without a calculator:

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

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Answer

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, $(a^2)^4 = a^{(24)}= a^8$ . 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, $(\frac{9y^2}{x^4})^\frac{3}{2} = \frac{9^{3/2}(y^2)^{3/2}}{(x^4)^{3/2}} = \frac{(\sqrt9)^3y^{(23/2)}}{x^{(4*3/2)}} = \frac{3^3y^3}{x^6}= \frac{27y^3}{x^6}$

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Question

Which of the following numbers is in scientific notation?

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Answer

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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Question

If , what does equal?

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Answer

We can use the fact that $(x^a)^b=x^{a*b}$ to see that

Since , we have

.

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Question

Rewrite as a single logarithmic expression:

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

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Answer

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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Question

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

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Answer

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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Question

Which of the following expressions is equal to the expression

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

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Answer

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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Question

Simplify:

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

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Answer

Use the properties of exponents as follows:

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

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

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

$=32 \cdot x^{6} \cdot x^{6}$

$=32 \cdot x^{6+6}$

$=32x^{12}$

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Question

Simplify:

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

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Answer

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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Question

Divide:

$\left (56x^{3} +21x ^{2} -63x + 77 \right )\div 7 x ^{2}$

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Answer

$\left (56x^{3} +21x ^{2} -63x + 77 \right )\div 7 x ^{2}$

$= \frac{ 56x^{3} }{7 x ^{2}} + \frac{ 21x ^{2} }{7 x ^{2}}- \frac{ 63x }{7 x ^{2}}+ \frac{ 77 }{7 x ^{2}}$

$= \frac{ 56 }{7 } x^{3-2} + \frac{ 21 }{7 }- \frac{ 63 }{7}\cdot \frac{1}{ x ^{2-1}}+ \frac{ 77 }{7} \cdot \frac{1}{ x ^{2}}$

$= 8 x +3 -\frac{9}{ x }+ \frac{ 11 }{x ^{2}}$

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Question

Solve for :

$\left (3 ^{4} \right )^{N}\cdot 3^{5} =\frac{1}{27}$

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Answer

$\left (3 ^{4} \right )^{N}\cdot 3^{5} =\frac{1}{27}$

$3 ^{4\cdot N} \cdot 3^{5} =3 ^{-3 }$

$3 ^{4\cdot N+5} =3 ^{-3 }$

The left and right sides of the equation have the same base, so we can equate the exponents and solve:

$4N\div 4 = -8 \div 4$

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Question

Solve for :

$\left (\frac{1}{25} \right )^{N} \cdot 5^{6} = 125$

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Answer

$\left (\frac{1}{25} \right )^{N} \cdot 5^{6} = 125$

$\left ( 5 ^{-2} \right )^{N} \cdot 5^{6} = 5^{3}$

$5 ^{-2 \cdot N } \cdot 5^{6} = 5^{3}$

$5 ^{-2 N +6 } = 5^{3}$

$-2N \div (-2 ) = -3 \div (-2 )$

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

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Question

Which of the following is equal to $\log 8 + \log 5 - \log 4$ ?

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Answer

$\log 8 + \log 5 - \log 4$

$= \log \left ( 8 \cdot 5 \right ) - \log 4$

$= \log 40 - \log 4$

$= \log \left ( 40 \div 4 \right )$

$= \log 10 = 1$

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Question

Which of the following is equal to $4 \log 3 - 2 \log 6$ ?

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Answer

$4 \log 3 - 2 \log 6$

$= \log\left ( 3^{4} \right ) - \log \left ( 6^{2} \right )$

$= \log 81 - \log 36$

$= \log\frac{ 81}{36}$

$= \log\frac{ 9}{4}$

$= \log 9 - \log 4$

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Question

Which of the following is true about $\log_{5} 1,000$ ?

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Answer

$5 ^{4} = 625; 5 ^{5} = 3,125$

Therefore,

$\log_{5} 625 = 4; \log_{5} 3,125 = 5$

$\log_{5} 625 <\log_{5} 1,000 < \log_{5} 3,125$

$4 <\log_{5} 1,000 < 5$

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Question

Which of the following is equal to $2 + \log 6$ ?

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Answer

$2 + \log 6 = \log \left ( 10^{2} \right ) + \log 6 = \log 100 + \log 6 = \log\left ( 100 \cdot 6 \right ) = \log 600$

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