Properties of exponents: rules with examples

Exponent notation provides a compact way to record repeated multiplication. For example, (3^5) means (3 \times 3 \times 3 \times 3 \times 3), which equals (243). The number (3) is the base, while (5) is the exponent or index.

The exponent laws make these expressions easier to manipulate without writing every repeated factor. They are especially useful in Year 10 and Year 11 maths classes, whether students are simplifying algebraic expressions, solving equations or preparing for an Australian senior pathway assessment.

Each rule has conditions. Some apply only when bases match, while negative and fractional exponents require extra care. Working through examples also helps distinguish an exponent from a coefficient and shows when parentheses change the base.

How exponent notation works

An exponent applies only to the base immediately beside it. In (x^3y^2), the base (x) has exponent (3), and (y) has exponent (2). The expression represents (x \times x \times x \times y \times y).

Coefficients and exponents have different jobs. In (5x^3), the coefficient is (5), while the base is (x). It means (5 \times x \times x \times x). By contrast, ((5x^3)^2) places the whole expression inside the brackets, so both (5) and (x^3) are raised to power (2).

The same idea applies when several factors are multiplied. For example,

[ 12x^2y \times x^3y^4 = 12x^{2+3}y^{1+4}=12x^5y^5. ]

The coefficients are multiplied, while matching bases have their exponents added.

Multiplying and dividing powers

When multiplying powers with the same base, keep the base and add the exponents:

[ a^m \times a^n=a^{m+n}. ]

For instance, (x^4 \times x^3=x^7), because the seven factors of (x) can be combined. This rule does not allow exponents to be added when the bases differ: (x^4y^3) cannot be reduced further.

For division, keep the matching base and subtract the exponents:

[ \frac{a^m}{a^n}=a^{m-n}. ]

Thus, (\frac{x^7}{x^2}=x^5). Any nonzero coefficient remains separate, so (\frac{6x^6}{2x^4}) simplifies to (3x^2).

Raising a power to another power multiplies the exponents: ((a^m)^n=a^{mn}). Therefore, ((x^2)^5=x^{10}). A power can also be distributed across factors inside brackets: ((ab)^n=a^nb^n). It cannot be distributed across addition, so ((x+y)^2) is not (x^2+y^2).

Zero, negative and fractional exponents

For any nonzero base, (a^0=1). This means (x^0=1), but the rule does not make (0^0) defined; zero to the power zero is generally left undefined in standard exponent laws.

A negative exponent places the base in a reciprocal:

[ a^{-n}=\frac{1}{a^n}. ]

Consequently, (5x^{-3}) becomes (\frac{5}{x^3}). Negative exponents are useful for measurements that shrink, while positive indices show repeated growth.

A fractional exponent represents a root. In (a^{m/n}), the denominator gives the root and the numerator gives the remaining power. For example,

[ 27^{2/3}=(\sqrt[3]{27})^2=3^2=9. ]

An Adelaide weather model might start at (30^\circ\text{C}) and multiply that temperature by (0.9) each hour. After two hours, (30(0.9)^2=24.3), illustrating how an exponent records repeated multiplication by the same factor.

Using the rules in expressions

Algebraic expressions may combine several laws, but each part should be handled separately. For example,

[ (3x^2y^{-1})^3=3^3x^6y^{-3} =\frac{27x^6}{y^3}. ]

The outer power applies to every factor, the exponents on (x) and (y) are multiplied by (3), and the negative exponent is converted into a reciprocal.

Exponents also appear in scientific notation, which uses powers of ten to represent large or small quantities. The road distance from Perth to Adelaide is roughly (2{,}130) kilometres, written as (2.13\times10^3) km. A mining shipment of 35 million tonnes can be written as (3.5\times10^7) tonnes without a long string of zeros.

Australian prices provide another application. An A$80 item on an Australian online marketplace becomes A$88 after 10% GST is added. If an amount grows by (5%) each month, repeated growth is written as (A$80(1.05)^3), which is approximately A$92.61.

A practical exponent routine

For revision, learn each law together with its restrictions rather than memorising isolated formulas.

A checking routine helps prevent mistakes, particularly when signs, fractions or coefficients appear.

Step-by-step solutions from CurryPlum can be used alongside these checks without replacing the written working. The practical takeaway is to identify each base, select the one law that applies, combine the indices carefully and then verify the simplified expression.