Laws of Exponents

The three core rules

Same base, multiplying: add the exponents: $x^m \cdot x^n = x^{m+n}$ . E.g. $2^4 \cdot 2^3 = 2^{7} = 128$ .
A power raised to a power: multiply the exponents: $(x^m)^n = x^{mn}$ . E.g. $(3^2)^5 = 3^{10} = 59049$ .
Negative exponent: take the reciprocal: $x^{-m} = \dfrac{1}{x^m}$ . E.g. $2^{-5} = \dfrac{1}{32}$ .

All the exponent rules in one table

The core three combine into the full set. Each holds for any non-zero base $x$ , $y$ :

Laws of exponents

Rule	Law	Example
Product	$x^m \cdot x^n = x^{m+n}$	$5^2 \cdot 5^3 = 5^5$
Quotient	$\dfrac{x^m}{x^n} = x^{m-n}$	$\dfrac{7^6}{7^4} = 7^2 = 49$
Power of a power	$(x^m)^n = x^{mn}$	$(2^3)^2 = 2^6 = 64$
Power of a product	$(xy)^n = x^n y^n$	$(2 \times 3)^2 = 4 \times 9 = 36$
Power of a quotient	$\left(\dfrac{x}{y}\right)^n = \dfrac{x^n}{y^n}$	$\left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}$
Zero exponent	$x^0 = 1$	$100^0 = 1$
Negative exponent	$x^{-m} = \dfrac{1}{x^m}$	$10^{-2} = \dfrac{1}{100}$
Fractional exponent	$x^{m/n} = \left(\sqrt[n]{x}\right)^m$	$27^{2/3} = 3^2 = 9$

The zero rule follows from the quotient rule: $\dfrac{x^m}{x^m} = x^{m-m} = x^0$ , and anything divided by itself is 1.

Worked example: a negative exponent on a fraction

Simplify $\left(\dfrac{-3}{4}\right)^{-3}$ .

Solution: A negative exponent on a fraction flips it, then applies the (now positive) power:

\left(\dfrac{-3}{4}\right)^{-3} = \left(\dfrac{4}{-3}\right)^3 = \left(\dfrac{-4}{3}\right)^3 = \dfrac{-64}{27}

Worked example: combining rules to avoid big numbers

Simplify $\dfrac{7^8 \times 12^8}{14^8 \times 6^8}$ without computing any of these to the 8th power directly.

Solution: Notice $7 \times 12 = 84 = 14 \times 6$ , the numerator and denominator are built from the same product, just split differently. Since $a^n b^n = (ab)^n$ :

\dfrac{7^8 \times 12^8}{14^8 \times 6^8} = \dfrac{(7\times12)^8}{(14\times6)^8} = \dfrac{84^8}{84^8} = 1

Worked example: rewriting in one base

Simplify $\dfrac{2^5 \times 8^2}{4^3}$ .

Solution: Write every number as a power of 2 ( $8 = 2^3$ , $4 = 2^2$ ), then use the product and quotient rules:

\dfrac{2^5 \times (2^3)^2}{(2^2)^3} = \dfrac{2^5 \times 2^6}{2^6} = 2^{5+6-6} = 2^5 = 32

Worked example: fractional and negative exponents

Evaluate $16^{-3/4}$ .

Solution: Deal with the minus sign as a reciprocal, then take the 4th root before cubing, which keeps the numbers small:

16^{-3/4} = \dfrac{1}{16^{3/4}} = \dfrac{1}{\left(\sqrt[4]{16}\right)^3} = \dfrac{1}{2^3} = \dfrac{1}{8}

Worked example: solving for an exponent

If $\left(\dfrac{5}{3}\right)^{-5} \times \left(\dfrac{5}{3}\right)^{11} = \left(\dfrac{5}{3}\right)^{8x}$ , find $x$ .

Solution: Same base on both sides, so just add the exponents on the left and match:

(-5) + 11 = 8x \;\Rightarrow\; 6 = 8x \;\Rightarrow\; x = \dfrac{6}{8} = \dfrac{3}{4}

The same idea solves $3^{x+1} = 81$ : write $81 = 3^4$ , so $x + 1 = 4$ and $x = 3$ .

Common mistakes

Multiplying exponents when multiplying powers. $x^2 \cdot x^3 = x^5$ , not $x^6$ . Exponents multiply only for a power of a power.
Using the rules across different bases. $2^3 \cdot 3^2$ can't be combined into one power. The product rule needs the same base.
Reading a negative exponent as a negative number. $2^{-3} = \dfrac{1}{8}$ , which is positive.
Mixing up $-2^2$ and $(-2)^2$ . $-2^2 = -4$ (square first, then the minus), while $(-2)^2 = 4$ .
Distributing a power over a sum. $(x + y)^2$ is $x^2 + 2xy + y^2$ , not $x^2 + y^2$ . The power-of-a-product rule only works for products.

Where this comes up in exams

The 2025-26 CBSE Class IX syllabus recalls the laws of exponents with integral powers in its Number Systems unit. Powers, exponents and logarithms are also listed in the General Aptitude syllabus that every GATE 2027 paper includes.

Checked against CBSE Mathematics curriculum, Classes IX-X (2025-26) and GATE 2027 information brochure (General Aptitude syllabus). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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