GRE Quant: Roots and Exponents

Exponents represent repeated multiplication of a number, and roots represent the inverse process of finding a base that produces a given value. Clear understanding of how powers and roots work builds ease with numerical expressions and supports accurate handling of many GRE Quant questions. Simple rules around exponents and roots lead to layered outcomes and help you track how values change within expressions involving powers and roots.

The following video, part of our GRE prep course online, explains roots and exponents in a clear and simple manner. It covers the basics, shows how these expressions behave under common operations, and then applies the concepts on GRE-style problems so you experience the application first-hand. Use this learning steadily across your GRE prep, practice exercises, GRE sectional mocks, and GRE mocks.

 

Basic Principles: Roots & Exponents (Surds & Indices)

Here are the basic principles for exponents and sign rules:

Exponent Rules

• Multiplication: A^m x Aⁿ = A^(m+n)
• Power of a Power: (A^m)ⁿ = A^mn
• Division: A^m / Aⁿ = A^(m-n)
• Negative Exponents: 1 / A^m = A^-m
• Reciprocal Rule: 1 / A^-m = A^m

Power of Sign Rules

• Negative Base with Even Exponent: (Negative)^Even = Positive
• Negative Base with Odd Exponent: (Negative)^Odd = Negative
• Positive Base with Even Exponent: (Positive)^Even = Positive
• Positive Base with Odd Exponent: (Positive)^Odd = Positive

These foundational rules are frequently used to solve complex problems and can be applied through various practice examples.

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Key Step: Turn the Bases into Prime Numbers

When solving exponential equations where the bases are different, the primary strategy is to convert all bases to their most basic prime form. This allows you to equate the exponents and solve for the unknown variable.

In the example provided, the prime base for all numbers involved is 3.

Step-by-Step Conversion

• 27 becomes (3)³
• 9 becomes (3)²
• 81 becomes (3)⁴

Mathematical Breakdown

The initial equation is: [(27 * 9)^½] / [81]^3/4 = 9^{(2 – x)}

1. Convert to base 3

[(3³ * 3³)^1/2] / [3⁴]^3/4 = (3²)^{(2 – x)}

2. Simplify the terms

• Inside the brackets: 3³ * 3² = 3⁵
• Numerator: (3⁵)^1/2 = 3^(5/2)
• Denominator: (3⁴)^3/4 = 3³
• Right side: (3²)^{(2 – x)} = 3^{(4 – 2x)}

3. Apply division rules for exponents

3^{(5/2 – 3)} = 3^{(4 – 2x)}

3^(-1/2) = 3^{(4 – 2x)}

4. Equate exponents and solve

-1/2 = 4 – 2x

2x = 4 + 1/2

2x = 9/2

x = 9/4

The value for x is 9/4.

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GRE-style Question

Correct Answer: A

Quantity A is greater.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

Quantity A

(16)⁶ / (8)^-3

Quantity B

(4)^-3 * (64)^-2 / (32)⁴

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

 

Explanation

Step 1: Convert Quantity A to base 2

• 16 is (2)⁴ and 8 is (2)³.
• The expression is (2⁴)⁶ / (2³)^-3.
• Multiply the exponents: 2²⁴ / 2^-9.
• When dividing, subtract the exponents: 2^{(24 – (-9))} = 2^{(24 + 9)} = (2)³³.

Step 2: Convert Quantity B to base 2

• 4 is (2)², 64 is (2)⁶, and 32 is (2)⁵.
• The expression is (2²)^-3 * (2⁶)^-2 / (2⁵)⁴.
• Multiply the exponents: (2)^-6 * (2)^-12 / (2)²⁰.
• Combine the top by adding exponents: 2^{(-6 + -12)} = (2)^-18.
• Divide by the bottom by subtracting exponents: 2^{(-18 – 20)} = (2)^-38.

Step 3: Compare results

• Quantity A is (2)³³.
• Quantity B is (2)^-38.
• Since 33 is much larger than -38, Quantity A is greater.

 

Correct Answer: A

Quantity A is greater.

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Concept: Cancel-out Big Exponents

When dealing with mathematical expressions involving large exponents, a common technique for simplification is to factor out the smallest power and cancel it from the numerator and denominator.

Problem Statement

The expression to solve is: (2¹⁰⁰ + 2¹⁰¹ + 2¹⁰² + 2¹⁰³) / (2⁹⁹ + 2¹⁰⁰ + 2¹⁰¹ + 2¹⁰²)

Solution Steps

• Identify the Common Factor: In this expression, (2)⁹⁹ is the smallest power present across both the top and bottom terms.
• Factoring Out: By taking 2⁹⁹ out of every term, the expression becomes: (2)⁹⁹ * (2¹ + 2² + 2³ + 2⁴) / (2)⁹⁹ * (2⁰ + 2¹ + 2² + 2³)
• Simplifying Inside Parentheses: This translates to: (2)⁹⁹ * (2 + 4 + 8 + 16) / (2)⁹⁹ * (1 + 2 + 4 + 8)
• Final Calculation: After canceling the (2)⁹⁹ from both sides, you are left with: 30 / 15 = 2

The core academic takeaway is that big exponents can be managed by identifying a common base and exponent to factor out, effectively reducing a complex fraction to simple integers.

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Conceptual Example

To simplify complex exponential expressions involving addition, the most effective strategy is to factor out the smallest common power. This allows the large exponents to cancel each other out, leaving behind manageable integers.

Problem Example

Evaluate (3^200 + 3^201 + 3^203) / (3^199 + 3^201 + 3^203)

Step 1: Identify and Factor the Lowest Power

In this expression, the lowest power among all terms is (3)¹⁹⁹. By factoring (3)¹⁹⁹ out of both the numerator and the denominator, the expression is rewritten as:

• Numerator: (3)¹⁹⁹ * (3¹ + 3² + 3⁴) which simplifies to (3)¹⁹⁹ * (3 + 9 + 81)
• Denominator: (3)¹⁹⁹ * (3⁰ + 3² + 3⁴) which simplifies to (3)¹⁹⁹ * (1 + 9 + 81)

Step 2: Cancel and Solve

The common factor of (3)¹⁹⁹ in the numerator and denominator cancels out completely. This leaves a simple fraction of the remaining sums:

• (3 + 9 + 81) / (1 + 9 + 81) = 93 / 91

The final simplified value of the expression is 93/91.

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Handle Negative Powers with Care…

Correct Answer: B

Quantity B is greater.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

x is an integer for which 1/(3)^-x is less than 1/81.

Quantity A

x

Quantity B

-4

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

 

Explanation

Step 1: Simplify the expression

The rule for negative exponents states that a negative power in the denominator becomes a positive power when moved to the numerator. Therefore, 1 / (3)^-x is equal to (3)^x.

Step 2: Rewrite the inequality

Now we can write the problem as: (3)^x < 1 / 81

Step 3: Convert 1 / 81 to a base of 3

We know that 81 = 3 * 3 * 3 * 3, which is (3)⁴. Using the rule for exponents, 1 / (3)⁴ is equal to (3)^-4. The inequality is now: (3)^x < (3)^-4

Step 4: Solve for x

Since the bases are the same and are greater than 1, we can compare the exponents directly: x < -4

Step 5: Compare Quantity A and Quantity B

Quantity A is x. Quantity B is -4. Because x must be less than -4 (such as -5, -6, etc.), Quantity B is always greater than Quantity A.

 

Correct Answer: B

Quantity B is greater.

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Powers Vs Bases…

Correct Answer: D

The relationship cannot be determined from the information given.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

p is a positive integer greater than 1

Quantity A

2^(p+2)

Quantity B

(3)^p

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

 

Explanation

Since p must be a positive integer greater than 1, we will test the smallest possible values for p.

Case 1: Let p = 2

Quantity A: 2⁽²⁺²⁾ = (2)⁴ = 16

Quantity B: (3)² = 9

In this case, Quantity A is greater than Quantity B.

Case 2: Let p = 3

Quantity A: 2⁽³⁺²⁾ = 2⁵ = 32
Quantity B: (3)³ = 27
In this case, Quantity A is still greater than Quantity B.

Case 3: Let p = 4

Quantity A: 2⁽⁴⁺²⁾ = (2)⁶ = 64
Quantity B: (3)⁴ = 81
In this case, Quantity B is greater than Quantity A.

Conclusion

Since Quantity A was greater in some cases and Quantity B was greater in another, the relationship cannot be determined.

 

Correct Answer: D

The relationship cannot be determined from the information given.

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Approximation…

Correct Answer: B

Quantity B is greater.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

Quantity A

√(15)/4 + √(8)/3

Quantity B

√(27)/5 + √(5)/2

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

 

Explanation

Step 1: Approximate Quantity A

• √(15) is slightly less than sqrt(16), which is 4. Let us call it 3.9.
• 3.9 / 4 is approximately 0.975.
• √(8) is slightly less than √(9), which is 3. Let us call it 2.8.
• 2.8 / 3 is approximately 0.933.
• Total for Quantity A: 0.975 + 0.933 = 1.908.

Step 2: Approximate Quantity B

• √(27) is slightly more than √(25), which is 5. Let us call it 5.2.
• 5.2 / 5 is 1.04.
• √(5) is slightly more than √(4), which is 2. Let us call it 2.2.
• 2.2 / 2 is 1.1.
• Total for Quantity B: 1.04 + 1.1 = 2.14.

Step 3: Compare

• Quantity A is about 1.91.
• Quantity B is about 2.14.
• Quantity B is clearly larger than Quantity A.

 

Correct Answer: B

Quantity B is greater.

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