Basics of Exponents | Quantitative Reasoning for GMAT PDF Download

Exponent Meaning

Exponent is a concise way to express repeated multiplication of the same number. In the expression 6⁴, the number 6 is the base and 4 is the exponent. This means 6 is multiplied by itself 4 times: 6 × 6 × 6 × 6. The expression is read as "6 raised to the power 4" or "6 to the power 4". Exponents allow large or repeated products to be written and manipulated compactly.

Exponent Symbol

The common ways to write an exponent are using a superscript, for example 4², or using the caret symbol ^, for example 4^2. Both represent the same value: 4² = 4 × 4 = 16.

Basic Laws of Exponents

The following laws apply for a non-zero base a and integers or rational exponents where the operations are defined. These are essential for simplifying expressions with powers.

• Product of like bases: a^m · aⁿ = a^m+n
• Quotient of like bases: a^m / aⁿ = a^m-n
• Power of a power: (a^m)ⁿ = a^mn
• Power of a product: (ab)ⁿ = aⁿ bⁿ
• Power of a quotient: (a/b)ⁿ = aⁿ / bⁿ
• Negative exponents: a^-k = 1 / a^k
• Zero exponent: a⁰ = 1 (provided a ≠ 0)
• One as exponent: a¹ = a

Exponent and Powers - Further Rules

Additional rules and clarifications that commonly appear in quantitative problems:

• Fractional exponents: a^1/n = n√a (the n-th root of a). More generally, a^m/n = (n√a)^m = (a^m)^1/n.
• Negative base with integer exponent: (-a)ⁿ depends on parity of n: if n is even, the result is positive; if n is odd, the result is negative. Parentheses matter: -a² = -(a²) but (-a)² = a².
• Zero base: 0ⁿ = 0 for n > 0; 0⁰ is indeterminate and avoided in standard algebraic manipulations.
• Distributivity caution: (a + b)ⁿ is not equal to aⁿ + bⁿ in general. Expansion requires the binomial theorem.

Common Mistakes and How to Avoid Them

• Confusing a^m · b^m with (ab)^m - these are equal, but a^m + b^m is not equal to (a + b)^m.
• Forgetting to change signs when dealing with negative exponents - a^-k = 1/a^k.
• Not converting bases to the same base when simplifying expressions like 9² with powers of 3; using equivalent base forms often simplifies calculations (for example 9 = 3²).
• Mixing parentheses: distinguish clearly between (-5)² and -5².

Exponent Table

A table of small bases and exponents is useful for quick reference and to check arithmetic when simplifying expressions involving powers.

Solved Examples

Example 1: Simplify (3² × 3^-5) / 9^-2

Ans: 
3² × 3^-5 = 3^{2 + (-5)}
3^{2 + (-5)} = 3^-3
Division by 9^-2 is the same as multiplication by 9².
So the expression becomes 3^-3 × 9²
Write 9 as 3², so 9² = (3²)²
(3²)² = 3⁴
Therefore 3^-3 × 3⁴ = 3^{-3 + 4}
3^{-3 + 4} = 3¹
Hence the value is 3.

Example 2: Simplify and write the answer in exponential form.

(i) ((2⁵ ÷ 2⁸)⁵) × 2^-5

Ans: 
Inside parentheses: 2⁵ ÷ 2⁸ = 2^5-8
2^5-8 = 2^-3
Raise to the 5th power: (2^-3)⁵ = 2^-15
Multiply by 2^-5: 2^-15 × 2^-5 = 2^-20
So the result is 2^-20 = 1 / 2²⁰.

(ii) (-4)^-3 × 5^-3 × (-5)^-3

Ans: 
All three factors have the same exponent -3, so combine: (-4 × 5 × -5)^-3
Compute inside: (-4 × 5) = -20
(-20 × -5) = 100
Therefore expression = 100^-3
100 = 10², so 100^-3 = (10²)^-3 = 10^-6
Hence the value is 100^-3 = 10^-6 = 1 / 1,000,000.

(iii) (1/8) × 3^-3

Ans: 
Write 1/8 as 2^-3 because 8 = 2³
So expression = 2^-3 × 3^-3
Combine using (ab)ⁿ = aⁿ bⁿ in reverse: 2^-3 × 3^-3 = (2 × 3)^-3
Therefore result = 6^-3.

Worked Example: Fractional Exponents

To relate fractional exponents to roots:

a^1/2 = √a
a^1/3 = ∛a
a^m/n = (n√a)^m = (a^m)^1/n
Example: 8^2/3 = (∛8)² = 2² = 4.

Applications (short notes relevant to quantitative problems)

• Compound growth or decay: Formulas such as A = P(1 + r/n)^nt use exponents to model repeated proportional change. Exponents capture the effect of repeated multiplication (compounding).
• Scientific notation: Large and small numbers are written as m × 10^k where m is a number (1 ≤ m < 10) and k is an integer. Understanding exponents is essential for working with scientific notation quickly and accurately.
• Complexity and scaling: In business or data problems, exponents appear when quantities grow multiplicatively (for example, user growth rates, interest problems, or repeated percentage changes).

Summary

Exponents are a compact notation for repeated multiplication. Mastery of exponent laws-product, quotient, power of a power, negative and fractional exponents-permits reliable simplification of algebraic expressions and accurate handling of growth/decay, scientific notation and many quantitative problems. Always pay attention to parentheses, convert to common bases when useful, and apply rules step by step to avoid sign and order errors.