Exponential Operations - GRE Quantitative Reasoning

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Question

Simplify

$\dpi{100} \small \frac{20x^{4}y^{-3}z^{2}}{5z^{-1}y^{2}x^{2}}=$

Answer

Divide the coefficients and subtract the exponents.

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Question

Which of the following is equal to the expression , where

xyz ≠ 0?

Answer

(xy)4 can be rewritten as x4y4 and z0 = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.

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Question

Quantitative Comparison: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.

Quantity A Quantity B

(23 )2 (22 )3

Answer

The two quantites are equal. To take the exponent of an exponent, the two exponents should be multiplied.

(23 )2 or 232 = 64

(22 )3 or 223 = 64

Both quantities equal 64, so the two quantities are equal.

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Question

Simplify: y3x4(yx3 + y2x2 + y15 + x22)

Answer

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

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Question

Indicate whether Quantity A or Quantity B is greater, or if they are equal, or if there is not enough information given to determine the relationship.

$\dpi{100} \small n>0$

Quantity A: $\dpi{100} \small 16^{n+2}$

Quantity B: $\dpi{100} \small 2^{4}\times (8^{n+1})^{2}\div 4^{n}$

Answer

By using exponent rules, we can simplify Quantity B.

$\dpi{100} \small \dpi{100} \small 2^{4}\times (8^{n+1})^{2}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times (8^{2n+2})\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times 2^{3(2n+2)}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times 2^{6n+6}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{6n+10}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{6n+10}\div 2^{2n}$

$\dpi{100} \small 2^{4n+10}$

Also, we can simplify Quantity A.

$\dpi{100} \small 16^{n+2}$

$\dpi{100} \small =2^{4(n+2)}$

$\dpi{100} \small =2^{4n+8}$

Since n is positive, $\dpi{100} \small 4n+10>4n+8$

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Question

If $5^2 \times 5^n = 5^{12}$ , what is the value of ?

Answer

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

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Question

If $\frac{3^{y - 1}}{3^{-2}} = 27^{y}3^{y}$ , what is the value of ?

Answer

Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).

$3^{y-1}3^2=27^y3^y$

The term on the right can be rewritten, as 27 is equal to 3 to the third power.

$3^{y-1}3^2=(3^3)^y3^y$

Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.

$3^{(y-1)+2}=(3)^{3y}3^y$

$3^{y+1}=3^{3y+y}=3^{4y}$

We now know that the exponents must be equal, and can solve for .

$\frac{1}{3}=y$

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Question

Simplify $x^{^{3}}x^{^{5}} - x^{^{2}} + (y^{^{2}})^{^{2}}$ .

Answer

First, simplify $x^{3}x^{5}$ by adding the exponents to get $x^{8}$ .

Then simplify $(y^{2})^{2}$ by multiplying the exponents to get $y^{4}$ .

This gives us $x^{8} - x^{^{2}} + y^{4}$ . We cannot simplify any further.

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Question

Simplify $x^{3}x^{6} + (x^{3})^{3} + x^{3}x^{2}x^{2}x^{2}$ .

Answer

Start by simplifying each individual term between the plus signs. We can add the exponents in and so each of those terms becomes . Then multiply the exponents in so that term also becomes . Thus, we have simplified the expression to which is $3x^{9}$ .

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Question

If $3^4 \cdot 3^8 = 9^x$ , what is the value of

Answer

To attempt this problem, note that .

Now note that when multiplying numbers, if the base is the same, we may add the exponents:

$3^4 \cdot 3^8 = 3^{4+8}=3^{12}$

This can in turn be written in terms of nine as follows (recall above)

$3^{12}=9^{\frac{12}{2}}=9^6$

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Question

If $(3^2\cdot3^5)^3=3^x$ , what is the value of

Answer

When dealing with exponenents, when multiplying two like bases together, add their exponents:

$3^2\cdot3^5=3^7$

However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:

$(3^7)^3=3^{7\cdot 3}=3^{21}$

$3^{21}=3^{x}$

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Question

If , then $\frac{(c^3)^2}{c^2c^4}=?$

Answer

Start by simplifying the numerator and denominator separately. In the numerator, (c3)2 is equal to c6. In the denominator, c2 * c4 equals c6 as well. Dividing the numerator by the denominator, c6/c6, gives an answer of 1, because the numerator and the denominator are the equivalent.

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Question

Evaluate:

$\frac{(2x^3y^{-2}a^6z^4)^5}{(4xy^4a^{-3}z^4)^2}$

Answer

Distribute the outside exponents first:

$\frac{2^5x^{15}y^{-10}a^{30}z^{20}}{4^2x^2y^8a^{-6}z^8}$

Divide the coefficient by subtracting the denominator exponents from the corresponding numerator exponents:

$\frac{2x^{13}a^{36}z^{12}}{y^{18}}$

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Question

If $a ot\equiv0$ , which of the following is equal to $\frac{(a^6*a^2)^3}{a^6}$ ?

Answer

The numerator is simplified to (by adding the exponents), then cube the result. a24/a6 can then be simplified to $a^{18}$ .

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Question

\[641/2 + (–8)1/3\] * \[43/16 – 3171/3169\] =

Answer

Let's look at the two parts of the multiplication separately. Remember that (–8)1/3 will be negative. Then 641/2 + (–8)1/3 = 8 – 2 = 6.

For the second part, we can cancel some exponents to make this much easier. 43/16 = 43/42 = 4. Similarly, 3171/3169 = 3171–169 = 32 = 9. So 43/16 – 3171/3169 = 4 – 9 = –5.

Together, \[641/2 + (–8)1/3\] * \[43/16 – 3171/3169\] = 6 * (–5) = –30.

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Question

$\dpi{100} \small \frac{7^{10}-7^{8}}{7^{9}-7^{7}}=$

Answer

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is .

$\frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}$

Now, we can cancel out the from the numerator and denominator and continue simplifying the expression.

$\frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}=\frac{7^3-7^1}{7^2-1}=\frac{343-7}{49-1}=\frac{336}{48}=7$

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Question

(b * b4 * b7)1/2/(b3 * bx) = b5

If b is not negative then x = ?

Answer

Simplifying the equation gives b6/(b3+x) = b5.

In order to satisfy this case, x must be equal to –2.

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Question

If〖7/8〗n= √(〖7/8〗5),then what is the value of n?

Answer

7/8 is being raised to the 5th power and to the 1/2 power at the same time. We multiply these to find n.

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Question

is a real number such that $y\geq 1$ .

Quantity A:

Quantity B:

Answer

(y2)(y4) = y2+4 = y6

Plug in two different values for y.

Plug in y = 1: y8 = y6

Plug in y = 2: y8 > y6

Since the results differ, the relationship cannot be determined from the information given.

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Question

Quantity A:

(0.5)3(0.5)3

Quantity B:

(0.5)7

Answer

When we have two identical numbers, each raised to an exponent, and multiplied together, we add the exponents together:

xaxb = xa+b

This means that (0.5)3(0.5)3 = (0.5)3+3 = (0.5)6

Because 0.5 is between 0 and 1, we know that when it is multipled by itself, it decreases in value. Example: 0.5 * 0.5 = 0.25. 0.5 * 0.5 * 0.5 = 0.125. Etc.

Thus, (0.5)6 > (0.5)7

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