Exponent: A number that shows how many times the base is multiplied by itself

General Form: \(a^n\) where \(a\) = base, \(n\) = exponent or power

Read as: "a to the power of n" or "a raised to the nth power"

Base: The number being multiplied

Exponent/Power: How many times to multiply the base

Example: \(5^3 = 5 \times 5 \times 5 = 125\)

Here, 5 is the base, 3 is the exponent, and 125 is the value

Exponential Form: \(2^5\)

Expanded Form: \(2 \times 2 \times 2 \times 2 \times 2\)

Standard Form: \(32\)

Step 1: Identify the base and exponent

Step 2: Multiply the base by itself the number of times indicated by the exponent

Step 3: Simplify to get the final answer

\(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

\(10^3 = 10 \times 10 \times 10 = 1000\)

\((-2)^3 = (-2) \times (-2) \times (-2) = -8\)

\(4^2 = 4 \times 4 = 16\)

Rule 1: If exponent is EVEN → Result is POSITIVE

Formula: \((-a)^{\text{even}} = +\text{(positive value)}\)

Example: \((-3)^2 = (-3) \times (-3) = +9\)

Rule 2: If exponent is ODD → Result is NEGATIVE

Formula: \((-a)^{\text{odd}} = -\text{(negative value)}\)

Example: \((-2)^3 = (-2) \times (-2) \times (-2) = -8\)

\((-5)^2 = (-5) \times (-5) = 25\) (negative base is squared)

\(-5^2 = -(5 \times 5) = -25\) (only 5 is squared, then negative sign applied)

Rule: Multiply the decimal by itself the number of times indicated by the exponent

Example: \((0.5)^2 = 0.5 \times 0.5 = 0.25\)

Example: \((0.1)^3 = 0.1 \times 0.1 \times 0.1 = 0.001\)

Formula: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Raise both numerator and denominator to the power

Example: \(\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}\)

Example: \(\left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1}{16}\)

Main Formula: \(a^{-n} = \frac{1}{a^n}\)

A negative exponent means "take the reciprocal and make the exponent positive"

Rule 1: \(a^{-n} = \frac{1}{a^n}\)

Example: \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)

Rule 2: \(\frac{1}{a^{-n}} = a^n\)

Example: \(\frac{1}{3^{-2}} = 3^2 = 9\)

Rule 3: \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\)

Example: \(\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^2 = \frac{25}{4}\)

Formula: \(a^0 = 1\) (where \(a \neq 0\))

Rule: Any non-zero number raised to the power of zero equals 1

\(5^0 = 1\)

\((-7)^0 = 1\)

\((100)^0 = 1\)

\(\left(\frac{3}{4}\right)^0 = 1\)

Note: \(0^0\) is undefined

Formula: \(a^m \times a^n = a^{m+n}\)

Rule: When multiplying powers with the same base, ADD the exponents

Integer Bases: \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\)

Variable Bases: \(x^5 \times x^2 = x^{5+2} = x^7\)

Negative Exponents: \(3^{-2} \times 3^5 = 3^{-2+5} = 3^3 = 27\)

Three or More: \(5^2 \times 5^3 \times 5^1 = 5^{2+3+1} = 5^6\)

Formula: \(\frac{a^m}{a^n} = a^{m-n}\)

Rule: When dividing powers with the same base, SUBTRACT the exponents

Integer Bases: \(\frac{5^7}{5^3} = 5^{7-3} = 5^4 = 625\)

Variable Bases: \(\frac{x^8}{x^3} = x^{8-3} = x^5\)

Negative Result: \(\frac{2^3}{2^5} = 2^{3-5} = 2^{-2} = \frac{1}{4}\)

Same Powers: \(\frac{7^4}{7^4} = 7^{4-4} = 7^0 = 1\)

Formula: \((a^m)^n = a^{m \times n}\)

Rule: When raising a power to another power, MULTIPLY the exponents

Integer Bases: \((3^2)^4 = 3^{2 \times 4} = 3^8 = 6561\)

Variable Bases: \((x^3)^5 = x^{3 \times 5} = x^{15}\)

Negative Exponents: \((2^{-3})^2 = 2^{-3 \times 2} = 2^{-6} = \frac{1}{64}\)

Complex: \((y^4)^{-2} = y^{4 \times (-2)} = y^{-8} = \frac{1}{y^8}\)

Formula: \((ab)^n = a^n \times b^n\)

Rule: When raising a product to a power, raise each factor to that power

\((2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36\)

\((xy)^4 = x^4 \times y^4 = x^4y^4\)

\((5ab)^3 = 5^3 \times a^3 \times b^3 = 125a^3b^3\)

Formula: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Rule: When raising a quotient to a power, raise both numerator and denominator to that power

\(\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}\)

\(\left(\frac{x}{y}\right)^5 = \frac{x^5}{y^5}\)

\(\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}\)

Step 1: Get the same base on both sides of the equation

Step 2: Set the exponents equal to each other

Step 3: Solve for the variable

Example 1: \(2^x = 8\)
Rewrite: \(2^x = 2^3\)
Solution: \(x = 3\)

Example 2: \(5^{x+1} = 25\)
Rewrite: \(5^{x+1} = 5^2\)
Solve: \(x + 1 = 2\)
Solution: \(x = 1\)

Example 3: \(3^{2x} = 81\)
Rewrite: \(3^{2x} = 3^4\)
Solve: \(2x = 4\)
Solution: \(x = 2\)

Step 1: Simplify inside parentheses

Step 2: Apply exponent rules (power of power, power of product, etc.)

Step 3: Evaluate powers

Step 4: Perform multiplication and division

Step 5: Perform addition and subtraction

Example 1: \(\frac{3^5 \times 3^2}{3^4}\)
Step 1: Multiply numerator → \(\frac{3^{5+2}}{3^4} = \frac{3^7}{3^4}\)
Step 2: Divide → \(3^{7-4} = 3^3 = 27\)

Example 2: \((2^3)^2 \times 2^{-4}\)
Step 1: Power of power → \(2^6 \times 2^{-4}\)
Step 2: Multiply → \(2^{6+(-4)} = 2^2 = 4\)

Example 3: \(\frac{(x^2y^3)^2}{x^3y^4}\)
Step 1: Power of product → \(\frac{x^4y^6}{x^3y^4}\)
Step 2: Divide → \(x^{4-3}y^{6-4} = xy^2\)

Method 1: Simplify both expressions and compare

Method 2: Use exponent properties to transform one expression into the other

Method 3: Substitute values and evaluate both expressions

\(x^3 \times x^2 \equiv x^5\)

\((2x)^3 \equiv 8x^3\)

\(\frac{x^6}{x^2} \equiv x^4\)

\((x^2)^3 \equiv x^6\)

\(x^{-2} \equiv \frac{1}{x^2}\)

❌ Mistake 1: \(x^2 \times x^3 \neq x^6\)

✓ Correct: \(x^2 \times x^3 = x^{2+3} = x^5\) (ADD exponents when multiplying)

❌ Mistake 2: \((x^2)^3 \neq x^5\)

✓ Correct: \((x^2)^3 = x^{2 \times 3} = x^6\) (MULTIPLY exponents for power of power)

❌ Mistake 3: \((2x)^3 \neq 2x^3\)

✓ Correct: \((2x)^3 = 2^3 \times x^3 = 8x^3\) (Apply exponent to ALL factors)

❌ Mistake 4: \(x^{-2} \neq -x^2\)

✓ Correct: \(x^{-2} = \frac{1}{x^2}\) (Negative exponent means reciprocal)

❌ Mistake 5: \(-3^2 \neq 9\)

✓ Correct: \(-3^2 = -(3^2) = -9\) but \((-3)^2 = 9\) (Watch parentheses!)

⚡ Remember: When MULTIPLYING → ADD exponents | When DIVIDING → SUBTRACT exponents | Power of Power → MULTIPLY exponents! ⚡

✓ Master the basics first: Understand what exponents mean before memorizing formulas

✓ Learn one property at a time: Don't try to memorize all rules at once

✓ Practice with numbers first: Use integers before moving to variables

✓ Check your work: Expand expressions to verify your answers

✓ Watch for parentheses: They completely change the meaning of expressions