Understanding exponents - GMAT Quantitative

Card 1 of 672

0

Didn't Know

0

Didn't Know

Knew It

0

1 of 2019 left

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

Divide:

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

Tap to reveal answer

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 }$+ $\frac{ 11 }{x ^{2}$}$

← Didn't Know|Knew It →

Question

. Order from least to greatest: .

Tap to reveal answer

Answer

If is negative, then:

Even powers and are positive, with and .

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

It follows that , and .

If , then - that is, . So changing the signs of the exponents reverses the order. As a result,

.

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

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

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

Setting the negative numbers less than the positive numbers:

.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

$3x^{4}$times $x^{2}$ $=3x^{6}$

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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

← Didn't Know|Knew It →

Question

If , what is in terms of ?

Tap to reveal answer

Answer

We have .

So , and .

← Didn't Know|Knew It →

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 $\small x > 0$ .

Tap to reveal answer

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}$

← Didn't Know|Knew It →

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}$$

Tap to reveal answer

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,

Doing this to our expression we get it simplified to .

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, . The other rule we must know that is an exponent of one half is the same as taking the square root. So for the So using these rules,

← Didn't Know|Knew It →

Question

Which of the following numbers is in scientific notation?

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

If , what does equal?

Tap to reveal answer

Answer

We can use the fact that to see that

Since , we have

.

← Didn't Know|Knew It →

Question

Solve for :

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

Tap to reveal answer

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:

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

← Didn't Know|Knew It →

Question

Rewrite as a single logarithmic expression:

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

Tap to reveal answer

Answer

First, write each expression as a base 3 logarithm:

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

$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$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

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

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

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.

← Didn't Know|Knew It →

Question

Which of the following expressions is equal to the expression

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

Tap to reveal answer

Answer

Use the properties of exponents as follows:

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

← Didn't Know|Knew It →

Question

Simplify:

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

Tap to reveal answer

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}$$

← Didn't Know|Knew It →

Question

Simplify:

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

Tap to reveal answer

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 }$

← Didn't Know|Knew It →

Question

Solve for :

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

Tap to reveal answer

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}$$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

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

$= \log 40 - \log 4$

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

$= \log 10 = 1$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$$

$= \log 9 - \log 4$

← Didn't Know|Knew It →

0

Knew It