What Are the Laws of Exponents With Integer Exponents?

The laws of exponents are rules for simplifying powers with integer exponents. They determine how exponents change when powers are multiplied, divided, or raised to another power, including zero and negative exponents.

The Laws

Let $m$ and $n$ be integers.

Why the Rules Work

The product and quotient laws come from counting repeated factors. For example,

$$\frac{x^5}{x^2}=\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}=x^3=x^{5-2}.$$

Similarly, $a^m/a^m=a^0=1$, provided $a\neq0$. Since , cancellation gives . A negative exponent therefore means a , not a negative value.

The power-of-a-power law also follows from repeated multiplication. For instance, $(a^3)^2$ means $a^3\times a^3$, so the product law gives . The original exponent is repeated twice, which explains why the exponents are multiplied rather than added. Restrictions such as matter because division by zero is undefined.

Applying the Laws

Separate coefficients from variables and apply one law at a time:

$$\frac{(2x^3)^2}{4x^{-1}}=\frac{4x^6}{4x^{-1}}=x^{6-(-1)}=x^7,$$

where $x\neq0$. Subtracting a negative exponent increases the exponent. You can check algebra by substituting a non-zero value into both the original and simplified expressions.

For another example,

$$\frac{6a^{-2}b^5}{3ab^{-1}}=2a^{-3}b^6=\frac{2b^6}{a^3}.$$

This final form uses positive exponents, as IB questions commonly require.

Common Misconceptions

Exponent laws do not apply across addition: $a^m+a^n\neq a^{m+n}$. For example, , not . Also, in general.

IB Exam Technique

This is SL 1.5, Exponents and Logarithms, studied in AI SL and HL. Show each law, simplify coefficients separately, state restrictions where relevant, and normally give answers with positive exponents. A GDC is available on every Math AI paper, but calculator output does not replace mathematical working.