19 June 2024

7 minutes read

# Exponent Rules GRE: GRE Quant Rules for Exponents

### Key Takeaways:

- Exponents are pervasive in GRE quant, so mastering the rules is non-negotiable for a top score.
- Add, subtract, multiply, divide – practice these operations with powers until they’re second nature.
- The “power of a power” rule (x^m)^n may seem tricky, but examples make it click – keep at it.
- Consistent practice with a variety of exponent-based questions is key to building lasting quant skills.

Are you preparing for the GRE and feeling overwhelmed by the exponent rules GRE? Don’t worry; you’re not alone. Many test-takers struggle with this concept, which can be confusing and tricky. Exponents are ubiquitous in the GRE Quantitative Reasoning section, making it crucial to master the rules governing them. GRE has huge scope, especially when it comes to pursuing courses like masters in engineering and so on.

In this comprehensive blog post, we’ll dive deep into the world of exponent rules for the GRE. We’ll break down the essential concepts, provide clear examples, documents needed for GRE exam, clear doubts regarding whether GRE has negative marking, and share valuable tips to help you conquer this challenging topic. Get ready to demystify exponents and boost your confidence in tackling even the most daunting GRE quantitative questions involving exponents.

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## Adding Powers

One of the fundamental rules in the realm of exponents is adding powers with the same base. This concept is highly relevant for GRE prep, as it frequently appears in quantitative reasoning questions. The rule states that when you have the same base raised to different exponents, you can add the exponents by keeping the base the same.

**Practice Questions:**

- Simplify: 3^4 x 3^2 Solution: 3^4 x 3^2 = 3^(4+2) = 3^6
- Evaluate: (2^3 x 2^5) / 2^2

Solution: (2^3 x 2^5) / 2^2 = 2^(3+5-2) = 2^6

As you can see from these examples, adding powers with the same base is a straightforward process that can simplify lengthy expressions. Mastering this rule is crucial for your GRE prep, as it can help you solve quantitative reasoning problems efficiently and accurately. You also have to keep at it by practicing frequently asked GRE math questions.

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## Subtracting Powers

Subtracting Powers with the Same Base (Paragraph) Another essential rule for working with exponents is subtracting powers with the same base. This concept is equally important for your GMAT and GRE prep. The rule states that when you have the same base raised to different exponents, you can subtract the exponents by keeping the base the same.

**Practice Questions:**

- Simplify: 5^8 / 5^3 Solution: 5^8 / 5^3 = 5^(8-3) = 5^5
- Factor: 8^x – 8^3 Solution: We can rewrite this as 8^3 (8^(x-3) – 1)

Subtracting powers with the same base allows you to simplify expressions and, in some cases, factor them into more manageable forms. Practicing this rule will undoubtedly benefit your GMAT and GRE test prep, enabling you to tackle quantitative reasoning questions with greater ease and confidence. The extent of GRE for your higher studies is huge, like you might need to take GRE for your masters as well.

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## Multiplying Powers

Multiplying powers with the same base is another fundamental rule you’ll encounter during your GRE test prep. This rule states that when you multiply two powers with the same base, you keep the base and add the exponents.

**Practice Questions:**

- Simplify: (2^3) x (2^5) Solution: (2^3) x (2^5) = 2^(3+5) = 2^8
- Evaluate: (x^4 * x^2) / (x^3) Solution: (x^4 * x^2) / (x^3) = x^(4+2-3) = x^3

Multiplying powers with the same base is a handy technique that can help you simplify expressions and solve problems more efficiently. By practicing this rule with GRE free practice tests and numerator /denominator questions, you’ll gain valuable experience and increase your chances of success on the actual GRE test.

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## Dividing Powers

Dividing powers with the same base is a crucial rule that millions of students need to know for their GRE and GMAT test prep. This rule states that when you divide one power by another power with the same base, you keep the base and subtract the exponents.

**Practice Questions:**

- Simplify: (x^8) / (x^5) Solution: (x^8) / (x^5) = x^(8-5) = x^3
- Factor out the greatest common factor: (3^x + 3^y) / 3^2

Solution: We can factor out 3^2 from the numerator: (3^x + 3^y) / 3^2 = 3^(x-2) + 3^(y-2)

Dividing powers with the same base is a valuable skill that can help you simplify expressions and solve problems more efficiently, especially when dealing with denominators. If you are still wondering whether you need to take GRE for your further studies, the answer is a solid yes, so start your prep soon. Depending on your capability and skills, you can crack GRE in one month even. There are also programs offering you to master GRE in two months as well, so choose your pace and start the prep.

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## Exponents with Exponents

One of the more advanced rules you’ll encounter in GRE problems and GMAT courses is exponents with exponents, also known as the “power of a power” rule. This rule applies when you have an expression like (x^m)^n, where both m and n are non-negative integers.

**Example:**

Evaluate: (2^3)^2 Solution: (2^3)^2 = 2^(3×2) = 2^6

Simplify: ((x^2)^3)^4 (Assume x is odd) Solution: ((x^2)^3)^4 = (x^6)^4 = x^24

Exponents with exponents can be tricky, and one of the common mistakes is confusing this rule with addition or subtraction involving powers. However, with proper math practice, you can master this concept and avoid such pitfalls. Understanding exponents with exponents is crucial for solving complex GRE problems and GMAT questions involving powers. You also can give coordinate geometry section of the GRE a try as it is an equally important section.

## GRE Exponent Practice Questions

Your career options after GRE is absolutely huge, so work hard during your prep stage to make the most out of the opportunity. If you are serious about your GRE exam, then the most important thing you have to do to crack the exam is practice. You can also join a good GRE online coaching class to help you with the prep, as well as crack into the best books available for GRE quant. Here are some questions to help you brush up your exponent skills:

**Q 1. If x^3 = 64, what is the value of x^(3/2)? **

Solution:

Step 1) x^3 = 64

Step 2) Taking the cube root of both sides: x = 64^(1/3) = 4

Step 3) x^(3/2) = 4^(3/2) = (4^3)^(1/2) = 64^(1/2) = 8

**Q 2. Simplify: (2^x * 3^y) / (2^(x-1) * 3^y)**

Solution:

Step 1) (2^x * 3^y) / (2^(x-1) * 3^y) = (2^x / 2^(x-1)) * (3^y / 3^y)

Step 2) 2^x / 2^(x-1) = 2^(x – (x-1)) = 2^1 = 2

Step 3) 3^y / 3^y = 1

Step 4) (2^x * 3^y) / (2^(x-1) * 3^y) = 2 * 1 = 2

**Q 3. If 2^(2x+1) = 32, what is the value of 2^(3x-2)? **

Solution:

Step 1) 2^(2x+1) = 32

Step 2) Taking log base 2 of both sides: (2x+1) = 5

Step 3) 2x = 4, so x = 2

Step 4) 2^(3x-2) = 2^(3(2)-2) = 2^4 = 16

**Q 4. Factor the expression: x^(n+2) – x^n **

Solution:

Step 1) x^(n+2) – x^n = x^n(x^2 – 1)

Step 2) Factor further: x^n(x^2 – 1) = x^n(x – 1)(x + 1)

**Q 5. If a^b = c and b^c = d, express d in terms of a and b. **

Solution:

Step 1) a^b = c

Step 2) Taking log base a of both sides: b = log_a(c)

Step 3) Substituting in b^c = d: (log_a(c))^c = d

Step 4) By the power rule: (log_a(c))^c = c^(log_a(c)) = a^(b*log_a(c)) = a^b

Step 5) Therefore, d = a^b

**Q 6. Solve for x: (2^x)^(1/3) = 4 **

Solution:

Step 1) (2^x)^(1/3) = 4

Step 2) Using the power of a power rule: 2^(x/3) = 4

Step 3) Taking log base 2 of both sides: x/3 = 2

Step 4) x = 6

**Q 7. If x^(1/2) = 2 and y^(1/3) = 4, what is the value of (xy)^(1/6)? **

Solution:

Step 1) x^(1/2) = 2, so x = 2^2 = 4

Step 2) y^(1/3) = 4, so y = 4^3 = 64

Step 3) (xy)^(1/6) = (4*64)^(1/6) = 68^(1/6)

Step 4) 68 = 4^3, so (68)^(1/6) = (4^3)^(1/6) = 4^(1/2) = 2

**Q 8. Simplify: (3^(x+2) * 3^x) / (3^(2x+1)) **

Solution:

Step 1) (3^(x+2) * 3^x) / (3^(2x+1)) = (3^(x+2+x)) / (3^(2x+1))

Step 2) = 3^(2x+2) / 3^(2x+1)

Step 3) = 3^(2x+2-2x-1) = 3^1 = 3

**Q 9. If 5^x = 25 and 5^y = 625, express 5^(x+y) in terms of 25 and 625.**

Solution:

Step 1) 5^x = 25, so x = 2

Step 2) 5^y = 625, so y = 4

Step 3) 5^(x+y) = 5^(2+4) = 5^6

Step 4) 5^6 = (5^2)^3 = 25^3 = 25 * 25 * 25 = 15,625

**Q 10. Evaluate: (x^(1/2) * y^(1/3))^6, given that x = 16 and y = 27. **

Solution:

Step 1) x = 16, so x^(1/2) = 16^(1/2) = 4

Step 2) y = 27, so y^(1/3) = 27^(1/3) = 3

Step 3) (x^(1/2) * y^(1/3))^6 = (4*3)^6 = 12^6 = (2^4 * 3^2)^6*

*Step 4) = 2^(4*6) * 3^(2*6) = 2^24 * 3^12

**Q 11. Factor out the greatest common factor: (2^(3x) – 2^(2x+1)) / (2^x)**

Solution:

Step 1) (2^(3x) – 2^(2x+1)) / (2^x)

Step 2) Factor out 2^x: 2^x(2^(2x) – 2^(x+1))

Step 3) Factor further: 2^x(2^x(2^x – 2))

**Q 12. If 2^(x-1) = 8, express 2^(x+2) in terms of 8.**

Solution:

Step 1) 2^(x-1) = 8

Step 2) Taking log base 2 of both sides: x – 1 = 3, so x = 4

Step 3) 2^(x+2) = 2^(4+2) = 2^6 = 8 * 4 = 32

**Q 13. Simplify: (a^(1/2) * b^(1/3)) / (a^(1/4) * b^(1/6))**

Solution:

Step 1) (a^(1/2) * b^(1/3)) / (a^(1/4) * b^(1/6))

Step 2) = (a^(1/2 – 1/4) * b^(1/3 – 1/6))

Step 3) = a^(1/4) * b^(1/6)

**Q 14. Solve for x: (3^x)^(1/2) = 9^(x-1)**

Solution:

Step 1) (3^x)^(1/2) = 9^(x-1)

Step 2) 3^(x/2) = 9^(x-1)

Step 3) Taking log base 3 of both sides: x/2 = (x-1)log_3(9)

Step 4) x/2 = (x-1)(2), so x = 4

**Q 15. If a^b = c^d and b^c = d^a, express a in terms of b, c, and d. **

Solution:

Step 1) a^b = c^d

Step 2) Taking log base c of both sides: (log_c(a))b = d

Step 3) b^c = d^a

Step 4) Taking log base d of both sides: c(log_d(b)) = a

Step 5) Substituting Step 2 into Step 4: c(log_d(b)) = (log_c(a))b

Step 6) Taking log base b of both sides: c(log_d(b)) / b = log_c(a)

Step 7) By the change of base formula: log_c(a) = (log_d(b) / b)log_c(d)

Step 8) Substituting Step 7 into Step 6: a = d^(log_d(b)/b)

Therefore, if a^b = c^d and b^c = d^a, then a = d^(log_d(b)/b).

## Conclusion

Mastering exponent rules is key for the GRE Quantitative Reasoning section. This blog equipped you with the fundamentals of adding, subtracting, multiplying, dividing, and working with exponents of exponents, along with practice questions and solutions. Consistent practice using these concepts will boost your quantitative skills. Refer back to this resource as you prepare, and approach exponent questions on the GRE with confidence. Dedication and perseverance will help you conquer this topic and achieve your target score. Make sure to clear the basic doubts regarding the exam as to whether you can take the GRE in December

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## FAQs

### What are the most important exponent rules to know for the GRE?

The fundamental rules you must master include adding, subtracting, multiplying, and dividing exponents with the same base, as well as working with exponents raised to a power (e.g., (x^m)^n).

### How can I practice applying exponent rules effectively?

Consistent practice with a variety of GRE-style questions involving exponents is crucial. Focus on understanding each step and identifying which rule(s) to apply in different scenarios.

### Are exponent rules only relevant for the Quantitative section?

While exponent rules are heavily emphasized in the Quant section, a strong grasp of these concepts can also benefit you in certain Verbal and Analytical Writing questions involving numeric patterns or sequences.

### What are some common mistakes to avoid when working with exponents?

Typical pitfalls include confusing the rules for addition/subtraction with multiplication/division of exponents, incorrectly applying the “power of a power” rule, and careless algebraic errors.

### How can I build my intuition for exponent-based questions?

In addition to practicing methodically, try to develop a conceptual understanding by relating exponents to real-world examples (e.g., scientific notation, compound interest, growth rates).

### Are there any memory aids or tricks for recalling exponent rules?

Creating memorable acronyms or mnemonics can be helpful (e.g., “APE” for Add Powers with same base = Exponent sum). Drawing diagrams or visualizing patterns can also aid retention.

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