GRE Exponents: Basics & Practice Problems

Exponents in the GRE refer to the mathematical operation where a base number is raised to a certain power, indicating how many times the base is multiplied by itself.  Exponents are considered a core component of the GRE quantitative reasoning section that includes topics such as adding/subtracting, multiplying/dividing, and raising exponents to exponents. GRE exponents typically account for 10-15% of the total questions on the GRE mathematics exam. Mastering the rules and operations of exponents is crucial, as they are used in GRE arithmetic, algebra, and problem-solving throughout the test. In this article, we'll cover GRE exponent basics, rules, and practice questions with answers. Let's get started!

Basics and Rules of GRE Exponents

Understanding the base is crucial in every GRE problem that includes powers. The base refers to the number beneath the exponent. For example, in 53, five is the base and three is the exponent. This notation signifies that you are multiplying 5 (the base) three times: 5×5×5=125.

Before exploring other exponent rules, it's essential to understand the basics of different types of exponents. 

Exponent Type	Definition
Squaring (n2)	Multiplying n by itself (n × n).
Cubing (n3)	Multiplying n by itself three times (n × n × n).
negative Exponents (n-k)	Represents 1/nk, where n is not zero.
Zero Exponent (n0)	Equals 1, provided n is not zero.

1. Adding Powers/Exponents

When you add similar bases in exponent form, you cannot simply add the powers. 

For instance:

3³+3⁵

You cannot add the exponents (3 and 5). Instead, you need to factorize:

3³+3⁵= 3³ (1+3²) = 3³ (1+9) = 3³ (10)

Remember, whether bases are the same or different, you cannot add their exponents directly.

Key Takeaways:

• Exponents cannot be added when the bases are added.
• The rule applies regardless of whether the bases are identical or different.
• To solve problems involving adding exponents, understanding how to factorize is essential.
• Always check for common bases or terms that can be factored out when dealing with exponents.

2. Subtracting Powers/Exponents

Everything about adding powers also applies to subtracting them. Remember, subtracting is essentially adding a negative number.

For example:

3⁵ - 3³

The answer is NOT 3²! Instead, you need to factorize:

3⁵ - 3³  = 3³ (3²-1) =  3³ (9-1) = 3³ (8) = 216

Key Takeaways:

• Exponents cannot be subtracted when you are subtracting the bases.
• This rule holds true whether the bases are the same or different.
• To solve problems involving the subtraction of exponents, factorization is necessary.

3. Multiplying Powers/Exponents

Consider 3³ × 3², The rule is: if the bases are the same and you are multiplying them, then you should add the powers. 

So, 3³ × 3² = 3 ³⁺² = 3⁵. 

Keep the base (3) the same and add the exponents.

Now, let's see for 3³ × 2². Since the bases are different, you cannot combine them like before. You would need to multiply the numbers directly: 

3³ × 2² = 27×4 =108.

There is an exception: if the exponents are the same but the bases are different, you can multiply the bases. For example, in 3³ × 2³, the bases are different but the exponents are the same. Here, you multiply the bases and keep the exponent: 3³ × 2³ = (3 × 2)³ = 6³  = 216.

Key Takeaways:

• Add powers when multiplying terms with the same bases.
• You can multiply exponential terms if the bases are the same or if the exponents/powers are the same.
• Remember always to check if the bases or exponents are the same before applying these exponent rules.

4. Dividing Powers/Exponents

Just like multiplying, dividing exponential terms follows a specific rule: when you’re dividing, you subtract the powers. 

For example, 3³ / 3² = 3^3-2 = 3¹ = 3. 

Keep the base (3) the same and subtract the exponents.

For 3³ / 2², since the bases are different, you cannot combine them. You would need to divide directly: that is, 3³ / 3²= 27/4 = 6.75

However, when the bases are different but the powers are the same, you can divide the bases. For example, in 4³ / 2³, the bases are different but the exponents are the same. Here, you divide the bases and keep the power: 4³ / 2³  = (4/2)³ = 2³.

Key Takeaways:

• Subtract the powers when dividing terms that have the same bases.
• You can divide exponential terms if the bases are the same or if the exponents/powers are the same.
• When bases are different, divide the numbers directly.

5. Exponents with Exponents

When dealing with exponents raised to other exponents, the base remains unchanged while the exponents are multiplied together. These are also called nested exponents, or power of a power, which occurs when an exponent is raised to another exponent. This means you have an exponentiation operation inside another exponentiation operation.

For example, (3⁴)²

In this scenario, you multiply the exponents/powers while keeping the base, 3, unchanged:

4 × 2 = 8

So, (3⁴)² = 3⁸

Key Takeaways:

• Maintain the same base when raising an exponent to another exponent.
• Multiply the exponents together.
• This rule simplifies expressions involving nested exponents.

GRE Exponent Practice Questions with Answers and Explanations

Below are GRE exponent practice questions with answers and explanations designed to help you strengthen your understanding of exponent rules.

Practice Questions on Adding & Subtracting Exponents

1. Reduce 2⁴ +  2⁶  to its simplest form.

Sol: Given: 2⁴ +  2⁶

Here, both terms have the same base 2 but different exponents (4 and 6). To combine these terms, follow these steps:

Factor out the common base raised to the smaller exponent:
 2⁴ +  2⁶ = 2⁴ × (1 +  2² )

Simplify inside the parentheses:
(1 +  2²) = 1 + 4 = 5

Multiply the result by 2⁴: 2⁴ × 5

Calculate 2⁴: 2⁴ = 16

Multiply 16 by 5 = 16 × 5 = 80

Therefore, 2⁴  +  2⁶ = 80

2. Solve  5² +  5²  +  5²

Sol: Given, 5² = 25

Since all terms have the same base 5 and the same exponent 2, you can add them directly by multiplying the base by the number of terms:
5² +  5² + 5² = 3 × 25 = 75

Therefore, 5²+ 5²+ 5²= 75.

3. If, 2^-y + 2^-y  = ⅛, what is the value of y?

Sol: Given, 2^-y + 2^-y = ⅛

Since both terms have the same base 2 and the same exponent -y, we can simplify this equation by adding the terms:
2 × 2^-y = ⅛

Now, divide both sides by 2:
2^-y = 1/16

To find y, we need to solve for -y:
−y = log₂​ (1/16​)
−y = -4

Therefore, the value of y = 4

4. Quantity A: 54 - 53 Quantity B: 625 − 125

(A) Quantity B is greater.
(B) Quantity A is greater.
(C)  The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Sol:  Let's calculate each quantity:

Quantity A: 5⁴ - 5³ = 625 - 125 = 500
Quantity B: 625 − 125 = 500

Since both Quantity A and Quantity B are equal, the two quantities are equal.

Therefore, the correct answer is (C): The two quantities are equal.

Practice Questions on Multiplying & Dividing Exponents

1. Reduce 64  ×  44   to its simplest form.

Sol: Let's reduce 6⁴  ×  4⁴ to its simplest form.

When multiplying terms with the same powers/exponents but different bases, you can multiply the bases and keep the exponent:

6⁴  ×  4⁴  = (6 × 4)⁴  = 24⁴

2. If, (9²)³ × 3⁵= 3^y , what is the value of y?

Sol: First, simplify (9²)³ :

 (9²)³  = 3 ^2(3×2)  = 3¹²

Now combine the exponents of 3:

3¹² × 3⁵ = 3¹²⁺⁵ = 3¹⁷

So, 3¹⁷ = 3y, which means y = 17.

3. If 5^-m ×  5 ^-m = 1/125, what is the value of m?

Sol: First, simplify the left side using the rules of exponents:

5^-m ×  5 ^-m = 5^(-m+(-m)) = 5^-m-m  = 5^-2m

Now we have:

5^-2m = 1/125

Next, express 1/125 as a power of 5:
125 = 5³
1/125 = 5^-3

So, we have:

5^-2m = 5^-3

Since the bases are the same, we can set the exponents equal to each other:

−2m = −3

Solving for m:
m = 3/2
m = 1.5

Therefore, the value of m is 1.5.

4. Find the sum of the digits of the integer z, where z = 6¹² × 4¹⁵

Sol: Simplify the Expression: 6¹² × 4¹⁵

Rewrite Using Powers of 2 and 3: 
6 = 2 x 3
4 = 2² 

So we can express the terms as:

6¹² = (2 x 3)¹² = 2)¹² x 3¹²
4¹⁵ = (2²)¹⁵ = 2³⁰

Combine the Expressions:

z = 2¹² x 3¹² x 2³⁰ = 2¹²⁺³⁰ x 3¹² = 2⁴² x 3¹²

Simplify Further, Notice that 2⁴² x 3¹² can be broken down:

2⁴² = (2 ³⁾¹⁴ = 8¹⁴
3¹² 

Forming the Number Using Powers of 10, We can express it as:

z = 8¹⁴   x 3¹²

Approximate the powers for simplicity:

8¹⁴  = (2³)¹⁴  = 2⁴² ≈ 4.398 × 10¹²
3¹² ≈ 531441

Multiply these approximations:

z ≈ 4.398 × 10¹² × 531441 ≈ 2.34×10¹⁸ 

This gives us a large number, so let's convert it to a simpler form for adding digits.

Calculate the Number and Sum the Digits:

Given, z ≈ 2.34×10¹⁸, this is a number followed by 18 zeros.

Therefore, summing the digits: 2+3+4=9

Thus, the sum of the digits of the integer z is 9.

5. Find the value of (78+ 77+76) /  75

Given:
7⁸ =  7⁵ × 7³ 
7⁷ =  7⁵ × 7²
7⁶ =  7⁵ × 7¹
7⁵ =  7⁵

Now, rewrite the expression:
(7⁸+ 7⁷+7⁶) /  7⁵ = (7⁵ × 7³) +  (7⁵ × 7²) +  (7⁵ × 7¹)  /  7⁵

Cancel out 7⁵ from the numerator and denominator:
= 7³  +  7² + 7

Now, calculate each term:
7³ = 343
7² = 49
7 = 7

Add them together:
343 + 49 + 7 = 399

Therefore, (7⁸+ 7⁷+7⁶) /  7⁵ = 399

6. Quantity A: (2³) × (2⁴) Quantity B: 2⁷

(A) Quantity B is greater.
(B) Quantity A is greater.
(C)  The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Sol: Let's solve each quantity:

Quantity A: (2³) × (2⁴) = 2³⁺⁴ = 2⁷

So, Quantity A simplifies to 2⁷
Quantity B: 2⁷

So, Quantity B is already 2⁷

Now, compare the results:
Quantity A: 2⁷
Quantity B: 2⁷

Since both Quantity A and Quantity B are equal, the two quantities are equal.

Therefore, the correct answer is (C): The two quantities are equal.

Practice Questions on Exponents Raised to Exponents

7. Reduce (4³ ×  4⁵)²   to its simplest form.

Sol: Given, (4³ ×  4⁵)² 

Combine the exponents inside the parentheses: 
4³ ×  4⁵ = 4³⁺⁵ = 4⁸

Now raise 4⁸  to the power of 2:
(4⁸)²  = 4^8x2 = 4¹⁶

Therefore, (4³ ×  4⁵) = 4¹⁶

8. Solve (3⁴)³  + (2³)²

Sol: Given, (3⁴)³  + (2³)²

First Calculate (3⁴)³
(3⁴)3 = 3^4x3 =  3¹²

Then calculate (2³)²
( 2³)² = 2^3x2  = 2⁶

Now, add these results together:
3¹²  + 2⁶.

9. Simplify (2³  ×  3² ×  4¹)² .

Sol: To simplify this expression, we'll apply the rule for exponents raised to exponents and simplify within the parentheses first.

Start with the given expression (2³  ×  3² ×  4¹)².
Simplify inside the parentheses:

2³  = 8
3²  = 9
4¹  = 4

So, (2³  ×  3² ×  4¹)  = 8 × 9 × 4

Calculate the product: 8 × 9 × 4 = 288
Therefore, (2³  ×  3² ×  4¹)  = 288

Now substitute back into the original expression: 
(2³  ×  3² ×  4¹)² = 288²

Apply the rule for exponents raised to exponents: 
288² = 288 × 288 = 82944

Therefore, (2³  ×  3² ×  4¹)²  = 82944

Additional Exponent Problems

10. If 0 < 5ⁿ < 10,000,  where n is a non-negative integer, what is the greatest value of 1/2ⁿ?
 
Sol: First, interpret the inequality 0 < 5ⁿ < 10,000:

5ⁿ < 10,000

Taking the logarithm base 5 on both sides gives:

n log_5​ (5) < log₅​ (10,000)

Since log₅​ (5) = 1 and log_5​ (10,000) = 4;

n<4

Therefore, n can take integer values from 0 to 3.

Now, let's evaluate 1/2ⁿ for each possible value of n:

For n=0, 1/2⁰ = 1
For n=1, 1/2¹ = ½ = 0.5
For n=2, 1/2² = 1/4 = 0.25
For n=3, 1/2³ = 1/8 = 0.125

Among these values, the greatest value of 1/2ⁿ occurs when n=0, which is 1.

Therefore, the greatest value of 1/2ⁿ where 0 < 5ⁿ < 10,000 is 1.

11. If n < 1/(-3)^(-n) ​< 500, where n is a prime number, how many unique integer values of n satisfy this inequality?

A) 1
B) 2
C) 3
D) 4
E) 5

Sol: Interpret the Inequality: n < 1/(-3)^(-n) ​< 500
Simplify 1/(-3)^(-n):
1/(-3)^(-n) = -3ⁿ
So the inequality becomes: n < −3ⁿ < 500

Test Values of n:

Start with the smallest prime number, which is 2:

• For n=2, 2 <−3² = −9 (not true)

Try the next prime number, 3: 

• For n=3, 3 <−3³ = −27 (not true)

Continue testing with the prime numbers:

• For n=5, 5 <−3⁵ = −243 (true, as -243 < 500)

From testing, it appears that n=5  is the only prime number that satisfies the inequality 
n < 1/(-3)^(-n) ​< 500

Final Answer: A) 1

12. Solve for x: 3^x = 81

A) 2
B) 3
C) 4
D) 5
E) 6

Sol: Recognize that 81 can be written as a power of 3:
81 = 3⁴
So, we have, 3^x = 3⁴

Since the bases are the same, we can set the exponents equal to each other:
x = 4

So, the correct answer is (C): 4

13. Solve for x: 3^x+1 = 243

A) 4
B) 5
C) 6
D) 7
E) 3

Sol: Recognize that 243 can be written as a power of 3:

So, we have, 3^x+1 = 3⁵

Since the bases are the same, we can set the exponents equal to each other:
x + 1 = 5

Subtract 1 from both sides: x = 4

So, the correct answer is (A): 4

14. Compare Quantity A and Quantity B. Use any additional information provided above the two quantities to help make the comparison.
Quantity A: 2⁵
Quantity B: 5²

A) The answer cannot be determined from the information given.
B) The two quantities are equal.
C) Quantity A is greater.
D) Quantity B is greater.
E) Both quantities cannot be compared with the given information.

Sol: First calculate Quantity A: 2⁵ = 32
Next, calculate Quantity B: 5² = 25

Comparing the two quantities:
32 (Quantity A) is greater than 25 (Quantity B).

So, the correct answer is: (C)
Quantity A is greater.

15. Which one of the following is different from the others?
A) 3¹⁵
B) 9^7.5
C) 27⁵
D) 81⁴
E) (1/3)^-15

Sol: Rewrite each option in terms of powers of 3:

A) 3¹⁵ is already in terms of base 3.
B) 9^7.5 = (32)7.5 = 3¹⁵
C) 27⁵ = (33)5 = 3¹⁵
D) 81⁴ = (34)4 = 3¹⁶
E) (1/3)^-15 = (3-1)^-15 = 3¹⁵

So, all options except 81⁴ are equal to 3¹⁵.

Thus, the correct answer is: (D) 81⁴.

16. Simplify 37 +  36 .

A) 3⁶ . 4
B) 3⁷ . 3
C) 3⁷
D) 3¹³ . 3
E) 3¹³

Sol: Factor out the common term 3⁶
3⁷ +  3⁶ = 3. 3⁶ +  3⁶
 = 3⁶ (3 + 1)
= 3⁶ . 4

Thus, the correct answer is: (A) 3⁶ . 4.

Frequently Asked Questions (FAQ’s)

1. Can you calculate exponents on a GRE calculator?

No, the GRE calculator does not have the capability to calculate exponents. It is a basic four-function calculator without features like exponents, squaring, or cubing. To find the result of an expression with exponents, you must simplify it manually before entering the calculation into the GRE calculator.

2. What are the 7 Laws/Rules of Exponents?

Here are the basic laws of exponents, which are fundamental to mastering algebra, as these rules enable the simplification and manipulation of algebraic expressions and equations:

1. Any number raised to the power of zero is equal to 1. x⁰ = 1
2. A negative exponent indicates the reciprocal of the number with a positive exponent. x^-m = 1/x^m
3. When multiplying two powers with the same base, add their exponents. x^m  x  xⁿ =  x^m+n
4. When dividing two powers with the same base, subtract the exponents. x^m/ xⁿ =  x^m-n
5. For powers with different bases but the same exponent, multiply the bases and keep the exponent unchanged. x^m x  y^m = (xy)^m
6. When a fraction is raised to an exponent, apply the exponent to both the numerator and the denominator. (x/y)^m  = x^m /  y^m
7. If a power is raised to another exponent, multiply the exponents. (x^m)ⁿ = x^mn