GMAT Focus Edition Online Course - Exponents and Roots

Building Blocks of Power: Demystifying Exponents for GMAT Mastery

After delving into the fundamental properties of integers, we now transition to exponents, a powerful mathematical notation that simplifies the representation of repeated multiplication. Exponents are ubiquitous on the GMAT, appearing in everything from basic arithmetic to complex algebraic expressions, number properties, and even geometry problems. A solid understanding of what exponents mean and how they function is not merely helpful; it is foundational. This lesson will meticulously break down the definition of exponents, their components, and their intuitive meaning, setting the stage for mastering the powerful rules that govern them.

I. Defining Exponents: The Notation of Repetition

An exponent is a shorthand way of writing repeated multiplication of the same number.

Formal Definition:

For any real number k and any positive integer n, the expression k^n (read as "k to the power of n" or "k to the n-th power") means k multiplied by itself n times.

In the expression k^n:

• k is called the base. It is the number being multiplied.

• n is called the exponent (or power or index). It indicates how many times the base is used as a factor in the multiplication.

Example Illustrations:

• 5^4: Here, 5 is the base, and 4 is the exponent.

  • It means 5 5 5 * 5.

  • The value is 625.

• x^3: Here, x is the base, and 3 is the exponent.

  • It means x x x.

Special Terminology for Common Exponents:

• Squared (Exponent of 2): When the exponent is 2, we say the base is "squared." This term comes from the formula for the area of a square (side * side = side^2).

  • 2^2 is read as "2 squared" or "2 to the power of 2." It means 2 * 2 = 4.

  • x^2 means x * x.

• Cubed (Exponent of 3): When the exponent is 3, we say the base is "cubed." This term comes from the formula for the volume of a cube (side side side = side^3).

  • 3^3 is read as "3 cubed" or "3 to the power of 3." It means 3 3 3 = 27.

  • y^3 means y y y.

II. The Meaning of Positive Integer Exponents: A Deeper Dive

The simple idea of "repeated multiplication" has several important implications depending on the base.

A. Positive Base:

When the base k is a positive number, k^n will always be positive. The value generally increases rapidly as n increases.

• 4^2 = 4 * 4 = 16

• 10^3 = 10 10 10 = 1,000

• (1/2)^3 = (1/2) (1/2) (1/2) = 1/8

B. Negative Base:

When the base k is a negative number, the sign of k^n depends on whether the exponent n is even or odd.

• If n is even, (-k)^n will be positive.

  • (-2)^2 = (-2) (-2) = 4 (Note: (-2)^2 is different from -2^2. The latter means -(22) = -4).

  • (-3)^4 = (-3) (-3) (-3) (-3) = 9 9 = 81

• If n is odd, (-k)^n will be negative.

  • (-2)^3 = (-2) (-2) (-2) = 4 * (-2) = -8

  • (-1)^5 = (-1) (-1) (-1) (-1) (-1) = -1

C. Base of 0 or 1:

• 0^n = 0 for any positive integer n. (e.g., 0^5 = 0 0 0 0 0 = 0).

  • Special Case: 0^0 is generally considered undefined or an indeterminate form in higher mathematics, but it is rarely tested on the GMAT in a way that requires its definition. Stick to positive integer exponents for the base 0.

• 1^n = 1 for any positive integer n. (e.g., 1^100 = 1).

III. Why Understanding Exponents Matters for the GMAT

• Foundation for Exponent Rules: A clear understanding of the definition makes the exponent rules (which will be covered in Lesson 3.2) intuitive rather than rote memorization.

• Number Properties: Many number properties questions involve exponents (e.g., "Is x^n even/odd/positive/negative?").

• Algebraic Simplification: Exponents are fundamental to simplifying algebraic expressions and solving equations.

• Data Sufficiency: Questions often involve variables raised to powers, requiring you to understand how the base and exponent affect the value and properties.

• Units Digit Problems: Finding the units digit of a large power relies on the cyclic nature of units digits, which stems from the definition of exponents.

• Problem-Solving Strategy: Recognizing exponential patterns can lead to efficient solutions.

IV. Common Pitfalls to Avoid on the GMAT Focus Edition

• Confusing Base and Exponent: Make sure you know which number is being multiplied and how many times.

• Incorrectly Handling Negative Bases: Remember the odd/even rule for exponents with negative bases. (-2)^2 = 4, but -2^2 = -4. The parentheses are crucial!

• Misinterpreting x^0 (covered in next lesson): While we're focusing on positive integer exponents now, this is a common trap.

• Assuming Large Exponent = Large Number: While generally true for bases greater than 1, fractions between 0 and 1 become smaller as the exponent increases (e.g., (1/2)^2 = 1/4, (1/2)^3 = 1/8).

Interactive Check Your Understanding:

1. Identify the base and exponent in 7^9.

2. Write (-4)^3 in expanded form and calculate its value.

3. What is the value of (1/3)^4?

4. Without calculating the full value, determine if (-5)^{10} is positive or negative.

Practice Questions:

1. Which of the following expressions is equivalent to (-2) (-2) (-2) * (-2)?

   a) -2^4

   b) (-2)^4

   c) - (2 * 4)

   d) 2^4

2. If x = -3, what is the value of x^3 - x^2?

3. Which of the following statements is true for all real numbers k and positive integers n?

   a) k^n is always positive.

   b) (-k)^n is always negative.

   c) If k is between 0 and 1, then k^n is greater than k.

   d) If k < 0 and n is even, then k^n > 0.

4. How many 3s are multiplied together to form the number 729?

   a) 3

   b) 4

   c) 5

   d) 6

5. Arrange the following values from smallest to largest: A = (-1)^5, B = (1/2)^2, C = (-2)^2, D = -3^2.

Solutions to Practice Questions:

1. Equivalent to (-2) (-2) (-2) * (-2):

   This is (-2) multiplied by itself 4 times.

   a) -2^4 = -(2 2 2 * 2) = -16 (Base is 2, then negate)

   b) (-2)^4 = (-2) (-2) (-2) * (-2) = 16 (Base is -2, then multiply)

   c) -(2 * 4) = -8

   d) 2^4 = 2 2 2 * 2 = 16

   The correct answer is b) (-2)^4. Note that b and d evaluate to the same value, but b is the direct representation of the given expanded form.

2. If x = -3, value of x^3 - x^2:

   x^3 = (-3)^3 = (-3) (-3) (-3) = 9 * (-3) = -27

   x^2 = (-3)^2 = (-3) * (-3) = 9

   x^3 - x^2 = -27 - 9 = -36.

3. Which statement is true for all real numbers k and positive integers n?

   a) k^n is always positive. (False, e.g., (-2)^3 = -8)

   b) (-k)^n is always negative. (False, e.g., (-2)^2 = 4)

   c) If k is between 0 and 1, then k^n is greater than k. (False, e.g., (1/2)^2 = 1/4, and 1/4 < 1/2)

   d) If k < 0 and n is even, then k^n > 0. (True. An even number of negative factors results in a positive product.)

   The correct answer is d) If k < 0 and n is even, then k^n > 0.

4. How many 3s are multiplied together to form 729?

   This asks for the exponent n such that 3^n = 729.

   • 3^1 = 3

   • 3^2 = 9

   • 3^3 = 27

   • 3^4 = 81

   • 3^5 = 243

   • 3^6 = 729 So, 6 3s are multiplied together. The correct answer is d) 6.

5. Arrange from smallest to largest: A = (-1)^5, B = (1/2)^2, C = (-2)^2, D = -3^2:

   • A = (-1)^5 = -1 (Negative base, odd exponent = negative result)

   • B = (1/2)^2 = 1/4 = 0.25 (Positive base, squared)

   • C = (-2)^2 = 4 (Negative base, even exponent = positive result)

   • D = -3^2 = -(3 * 3) = -9 (Exponent applies only to 3, then negate)

   Arranging from smallest to largest:

   -9 < -1 < 0.25 < 4

   So, D < A < B < C.