Important Formulas: Exponents & Powers

Base is the number (or algebraic expression) that is multiplied repeatedly. Exponent (or power) is the small number written above and to the right of the base which shows how many times the base is used as a factor.

If a is a non-zero number and n is a positive integer then aⁿ means a multiplied by itself n times.

Examples:

• 2³ = 2 × 2 × 2 = 8
• 5¹ = 5
• a⁴ = a × a × a × a

Laws of Exponents

• \(a^{m} \times a^{n} = a^{m+n}\)

  When multiplying two powers with the same base, add the exponents. Valid for any integers \(m,n\) and for any base \(a\).

• \(\dfrac{a^{m}}{a^{n}} = a^{m-n}\)

  When dividing powers with the same base, subtract the exponents. Valid for \( a \neq 0 \).

• \((a^{m})^{n} = a^{mn}\)

  When a power is raised to another power, multiply the exponents.

• \((ab)^{m} = a^{m} b^{m}\)

  The power of a product equals the product of the powers. Valid for any integers \(m\).

• \(\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}\)

  The power of a quotient equals the quotient of the powers. Valid for \( b \neq 0 \).

• \(a^{-m} = \dfrac{1}{a^{m}}\)

  A negative exponent denotes reciprocal of the positive exponent. Valid for \( a \neq 0 \).

• \(a^{0} = 1\)

  Any non-zero number raised to the zero power equals 1.

• \(\left(\dfrac{a}{b}\right)^{-m} = \left(\dfrac{b}{a}\right)^{m}\)

  Negative exponent on a fraction swaps numerator and denominator, then applies the positive exponent. Valid for \( a \neq 0 \) and \( b \neq 0 \).

• \(1^{n} = 1\)

  One raised to any integer power is 1.

• \((-1)^{p} = 1\)

  When \(p\) is an even integer, \((-1)^{p} = 1\). When \(p\) is an odd integer, \((-1)^{p} = -1\).

Additional Remarks 

• Zero base: For positive integer \(n\), 0ⁿ = 0.
• Undefined form:0⁰ is indeterminate/undefined in elementary arithmetic and should be avoided.
• Negative exponents require non-zero base: Expressions like 0^-1 are not defined.
• Fractional exponents: These represent roots (for example, \(a^{1/2}=\sqrt{a}\)). This extends the laws but is treated under rational exponents (covered in higher classes).

Reasoning and Short Proofs

Product rule:

\(a^{m} = \underbrace{a \times a \times \cdots \times a}_{m\ \text{times}}\)

\(a^{n} = \underbrace{a \times a \times \cdots \times a}_{n\ \text{times}}\)

Multiplying these gives \(a^{m} \times a^{n} = \underbrace{a \times \cdots \times a}_{m+n\ \text{times}} = a^{m+n}\).

Quotient rule:

\(a^{m} = \underbrace{a \times \cdots \times a}_{m\ \text{times}}\)

\(a^{n} = \underbrace{a \times \cdots \times a}_{n\ \text{times}}\)

Dividing \(a^{m}\) by \(a^{n}\) cancels common factors and leaves \(a^{m-n}\) when \(m \ge n\).

Power of a power:

\((a^{m})^{n} = \underbrace{a^{m} \times a^{m} \times \cdots \times a^{m}}_{n\ \text{times}}\)

Each \(a^{m}\) contributes \(m\) factors of \(a\); total factors \(= mn\). Thus \((a^{m})^{n}=a^{mn}.\)

Negative exponent:

\(a^{m} \times a^{-m} = a^{m-m} = a^{0} = 1\)

Hence \(a^{-m} = 1/a^{m}\) for \( a \neq 0 \).

Zero exponent:

From the quotient rule with \(m=n\): \(\dfrac{a^{m}}{a^{m}} = a^{m-m} = a^{0}\).

But \(\dfrac{a^{m}}{a^{m}} = 1\) when \( a \neq 0 \). Thus \(a^{0} = 1\) for \( a \neq 0 \) and  \( m = 0 \).

Solved Examples

Example 1. Simplify \(a^{3} \times a^{4}\).

Sol: Use the product rule.

\(a^{3} \times a^{4} = a^{3+4}\)

\(= a^{7}\)

Example 2. Simplify \(2^{3} \times 2^{-5}\).

Sol: Apply the product rule (add exponents).

\(2^{3} \times 2^{-5} = 2^{3+(-5)}\)

\(= 2^{-2}\)

\(= \dfrac{1}{2^{2}}\)

\(= \dfrac{1}{4}\)

Example 3. Simplify \((3^{2})^{4}\).

Sol: Use power of a power (multiply exponents).

\((3^{2})^{4} = 3^{2\times 4}\)

\(= 3^{8}\)

Example 4. Simplify \((2\times 3)^{2}\).

Sol: Use power of a product.

\((2\times 3)^{2} = 2^{2} \times 3^{2}\)

\(= 4 \times 9\)

\(= 36\)

Example 5. Simplify \(\left(\dfrac{5}{2}\right)^{-3}\).

Sol: Apply the negative exponent rule to the fraction.

\(\left(\dfrac{5}{2}\right)^{-3} = \left(\dfrac{2}{5}\right)^{3}\)

\(= \dfrac{2^{3}}{5^{3}}\)

\(= \dfrac{8}{125}\)

Practice Questions

• Simplify \(x^{5} \times x^{-2}\).
• Simplify \(\dfrac{7^{4}}{7^{2}}\).
• Simplify \((xy)^{3}\).
• Simplify \(\left(\dfrac{4}{3}\right)^{2} \times \left(\dfrac{3}{4}\right)^{-1}\).
• Evaluate \((-1)^{6}\) and \((-1)^{7}\).