GRE Math : Algebra

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Example Questions

Example Question #2 : Exponential Operations

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

Possible Answers:

$\dpi{100} \small 49$

$\dpi{100} \small 7$

$\dpi{100} \small 343$

$\dpi{100} \small 28$

$\dpi{100} \small 42$

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 #42 : Exponents

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

Possible Answers:

y³x¹² + y⁶x⁸ + y⁴⁵ + x⁸⁸

y³x¹² + y¹²x⁸ + y²⁴x⁴ + y³x²³

y⁴x⁷ + y⁵x⁶ + y¹⁸x⁴ + y³x²⁶

y³x¹² + y⁶x⁸ + y⁴⁵x⁴ + y³x⁸⁸

2x⁴y⁴ + 7y¹⁵ + 7x²²

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 #3 : 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 A is greater.

The relationship cannot be determined from the information given.

The quantities are equal.

Quantity B 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 #2 : Exponential Operations

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

Possible Answers:

$\frac{3}{2}$

$\frac{1}{3}$

$\frac{2}{3}$

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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Example Question #43 : Exponents

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #53 : Exponents

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

Possible Answers:

$3x^{3}$

$x^{18} + 2x^{9}$

$2x^{9} + x^{27}$

$x^{27} + x^{18} + x^{9}$

$3x^{9}$

Correct answer:

$3x^{9}$

Explanation:

Start by simplifying each individual term between the plus signs. We can add the exponents in x^3x^6 and x^3x^2x^2x^2 so each of those terms becomes x^9 . Then multiply the exponents in (x^3)^3 so that term also becomes x^9 . Thus, we have simplified the expression to x^9 + x^9 + x^9 which is $3x^{9}$ .

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Example Question #451 : Algebra

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

Possible Answers:

$x^{15} - x^{2} + y^{4}$

$(xy)^{2}$

$x^{8} - x^{2} + y^{2}$

$x^{8} - x^{^{2}} + y^{4}$

$x^{6}+y^{4}$

Correct answer:

$x^{8} - x^{^{2}} + y^{4}$

Explanation:

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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Example Question #6 : How To Add Exponents

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

Possible Answers:

$\frac{32}{9}$

$\frac{4}{3}$

Correct answer:

Explanation:

To attempt this problem, note that 3^2=(3)^2=9^1=9 .

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)

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

9^6=9^x

x=6

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Example Question #452 : Algebra

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

Possible Answers:

Correct answer:

Explanation:

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:

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

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

x=21

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Example Question #1 : How To Subtract Exponents

Simplify: 3² * (4²³ - 4²¹)

Possible Answers:

3^21

3^3 * 4^21 * 5

4^4

3^3 * 4^21

None of the other answers

Correct answer:

3^3 * 4^21 * 5

Explanation:

Begin by noting that the group (4²³ - 4²¹) has a common factor, namely 4²¹. You can treat this like any other constant or variable and factor it out. That would give you: 4²¹(4² - 1). Therefore, we know that:

3² * (4²³ - 4²¹) = 3² * 4²¹(4² - 1)

Now, 4² - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:

3² * 4²¹(4² - 1) = 3² * 4²¹(3 * 5)

Combining multiples withe same base, you get:

3³ * 4²¹ * 5

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GRE Math : Algebra

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #2 : Exponential Operations

Example Question #42 : Exponents

Example Question #3 : Exponential Operations

Example Question #2 : Exponential Operations

Example Question #43 : Exponents

Example Question #53 : Exponents

Example Question #451 : Algebra

Example Question #6 : How To Add Exponents

Example Question #452 : Algebra

Example Question #1 : How To Subtract Exponents

All GRE Math Resources

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