GRE Math : Exponential Operations

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Example Question #1 : How To Divide Exponents

Simplify

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

Possible Answers:

None

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

$\dpi{100} \small {4x^{5}y^{-2}}$

$\dpi{100} \small 15x^{-2}y^{-2}z^{-2}$

$\dpi{100} \small 15x^{2}y^{2}z^{2}$

Correct answer:

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

Explanation:

Divide the coefficients and subtract the exponents.

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Example Question #1 : How To Divide Exponents

Which of the following is equal to the expression Equationgre , where

xyz ≠ 0?

Possible Answers:

1/y

xyz

z/(xy)

Correct answer:

1/y

Explanation:

(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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Example Question #1 : How To Divide Exponents

If $c\neq 0$ , then $\frac{(c^3)^2}{c^2c^4}=?$

Possible Answers:

c^4

c^2

Cannot be determined

Correct answer:

Explanation:

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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Example Question #1 : How To Divide Exponents

If $a\not\equiv0$ , which of the following is equal to $\frac{(a^6*a^2)^3}{a^6}$ ?

Possible Answers:

a⁶

The answer cannot be determined from the above information

a⁴

a¹⁸

Correct answer:

a¹⁸

Explanation:

The numerator is simplified to a^8 (by adding the exponents), then cube the result. a²⁴/a⁶ can then be simplified to $a^{18}$ .

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Example Question #1 : How To Divide Exponents

[64^1/2 + (–8)^1/3] * [4³/16 – 3¹⁷¹/3¹⁶⁹] =

Possible Answers:

–30

–5

Correct answer:

–30

Explanation:

Let's look at the two parts of the multiplication separately. Remember that (–8)^1/3 will be negative. Then 64^1/2 + (–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 = 3^171–169 = 3² = 9. So 4³/16 – 3¹⁷¹/3¹⁶⁹= 4 – 9 = –5.

Together, [64^1/2 + (–8)^1/3] * [4³/16 – 3¹⁷¹/3¹⁶⁹] = 6 * (–5) = –30.

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Example Question #1 : Exponential Operations

Evaluate:

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

Possible Answers:

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

$\frac{5x^{13}y-2az^{12}}{4}$

$\frac{5x^{13}y-2az^{12}}{4y}$

$2x^{13}-18a-36z^{12}$

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

Correct answer:

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

Explanation:

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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Example Question #151 : Exponents

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

Possible Answers:

$\dpi{100} \small 28$

$\dpi{100} \small 42$

$\dpi{100} \small 7$

$\dpi{100} \small 343$

$\dpi{100} \small 49$

Correct answer:

$\dpi{100} \small 7$

Explanation:

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 7^7 .

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

Now, we can cancel out the 7^7 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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Example Question #4 : How To Add Exponents

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

Possible Answers:

2x⁴y⁴ + 7y¹⁵ + 7x²²

y⁴x⁷ + y⁵x⁶ + y¹⁸x⁴ + y³x²⁶

y³x¹² + y⁶x⁸ + y⁴⁵ + x⁸⁸

y³x¹² + y¹²x⁸ + y²⁴x⁴ + y³x²³

y³x¹² + y⁶x⁸ + y⁴⁵x⁴ + y³x⁸⁸

Correct answer:

y⁴x⁷ + y⁵x⁶ + y¹⁸x⁴ + y³x²⁶

Explanation:

When you multiply exponents, you add the common bases:

y⁴ x⁷ + y⁵x⁶ + y¹⁸x⁴ + y³x²⁶

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Example Question #1 : Exponential Operations

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

Possible Answers:

Quantity B is greater.

The relationship cannot be determined from the information given.

The quantities are equal.

Quantity A is greater.

Correct answer:

Quantity B is greater.

Explanation:

By using exponent rules, we can simplify Quantity B.

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

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

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

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

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

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

Also, we can simplify Quantity A.

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

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

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

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Example Question #6 : Exponential Operations

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

Possible Answers:

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

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.

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

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 .

y+1=4y

1=3y

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

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GRE Math : Exponential Operations

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : Exponential Operations

Example Question #151 : Exponents

Example Question #4 : How To Add Exponents

Example Question #1 : Exponential Operations

Example Question #6 : Exponential Operations

All GRE Math Resources

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