Laws of exponents

Writing $2\times2\times2\times2\times2$ every time you need "two multiplied by itself five times" gets clumsy fast — so mathematicians shortened it to $2^5$ . Once numbers are written as powers, a second question appears: how do you multiply, divide or compare powers without expanding them back into long strings of multiplication? The laws of exponents answer exactly that. They are a small set of shortcut rules — seven in all — that let you combine powers with the same base directly, handle a power raised to another power, share an exponent across a product or a quotient, and make sense of a zero or negative exponent. This chapter of ICSE Class 8 Mathematics builds on the "power of a number" idea from earlier classes and extends it to negative integer exponents, one of the most-used toolkits in all of algebra.

The core insight

Powers with the same base combine by adding or subtracting their exponents on multiplication or division, by multiplying exponents when a power is raised to a power, and a negative exponent always means "take the reciprocal," never "make the value negative."

The core idea

The laws of exponents govern how powers behave for a nonzero base $a$ and integers $m, n$ . There are three families to keep straight.

Combining powers of the same base. Multiplying keeps the base and adds the exponents; dividing keeps the base and subtracts them.

a^m \cdot a^n = a^{m+n} \qquad\text{and}\qquad a^m \div a^n = a^{m-n}

where: $a$ is the base (the number being repeatedly multiplied, $a \neq 0$ ), and $m, n$ are the exponents (or indices) — integers that can be positive, negative or zero.

Distributing a single exponent. A power raised to another power multiplies the exponents; an exponent on a product or a quotient applies to each factor separately.

(a^m)^n = a^{mn} \qquad (ab)^n = a^n b^n \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Zero and negative exponents. These two are definitions built to keep the pattern above consistent, not new ideas from repeated multiplication.

a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad (a \neq 0)

The one rule to internalise before anything else: the product and quotient laws only add or subtract exponents when the bases are identical. $2^3 \times 3^2$ cannot be squashed into a single power of $2$ or of $3$ — there is nothing to combine. A negative exponent is not a sign flip on the value; $a^{-n}$ simply relocates the power to the denominator of a fraction. The figure below shows why the product law works, tile by tile.

Figure

Two rows of unit tiles for the powers of two showing three tiles beside two tiles joining into one row of five tiles to illustrate that multiplying powers adds their exponents

The core idea continued

Two of the seven laws deserve a closer look because they are definitions, not observations.

Why $a^0 = 1$ . Apply the division law to $a^m \div a^m$ . The left side is a number divided by itself, which is $1$ ; the right side, by the rule $a^m \div a^n = a^{m-n}$ , is $a^{m-m} = a^0$ . For both sides to agree, $a^0$ must equal $1$ — for every nonzero $a$ .

Why $a^{-n} = \dfrac{1}{a^n}$ . Apply the same division law to $a^0 \div a^n$ . The right side gives $a^{0-n} = a^{-n}$ . But $a^0 \div a^n$ is also just $1 \div a^n = \dfrac{1}{a^n}$ by ordinary division. Matching the two expressions gives $a^{-n} = \dfrac{1}{a^n}$ : the negative sign is an instruction to reciprocate, produced entirely by the pattern of the division law — no new arithmetic idea is needed.

Derivation ADeriving the negative-exponent law from the division law

The goal is to show, using only the division law $a^m \div a^n = a^{m-n}$ that is already accepted for positive exponents, that a negative exponent $a^{-n}$ must mean the reciprocal $\dfrac{1}{a^n}$ — turning what looks like a separate rule into a direct, checkable consequence of one law applied consistently, for any nonzero base $a$ and positive integer $n$ .

Step 1. Start from a division that is easy to evaluate two different ways: $a^n \div a^{2n}$ , where $n$ is a positive integer. Written out as a fraction, $a^n \div a^{2n} = \dfrac{a^n}{a^{2n}}$ , and cancelling $n$ matching factors of $a$ from top and bottom leaves $\dfrac{1}{a^n}$ in the denominator, since $2n - n = n$ factors of $a$ remain uncancelled below.

Step 2. Now evaluate the same division using the division law directly, without cancelling anything by hand: $a^n \div a^{2n} = a^{n - 2n} = a^{-n}$ . The law simply subtracts the exponents, giving a single power with a negative index.

Step 3. Compare the two results for the same expression. Step 1 gave $\dfrac{1}{a^n}$ by ordinary cancellation; Step 2 gave $a^{-n}$ by the division law. Since both computed the identical quantity $a^n \div a^{2n}$ , they must be equal: $a^{-n} = \dfrac{1}{a^n}$ .

Step 4. Check the pattern holds for a concrete case. With $a = 2$ and $n = 3$ : $2^3 \div 2^6 = 8 \div 64 = \dfrac{1}{8}$ by direct cancellation, and $2^{3-6} = 2^{-3}$ by the law — and indeed $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$ . The two routes agree, confirming that $a^{-n} = \dfrac{1}{a^n}$ is forced by the division law, not an extra rule bolted on.

Worked example

Simplify: $\dfrac{2^5 \times 2^{-2} \times 3^4}{2^3 \times 3^6}$

Step 1. Group the same bases together. Separate the base-2 terms from the base-3 terms so each family can be combined on its own:

\frac{2^5 \times 2^{-2}}{2^3} \times \frac{3^4}{3^6}

Step 2. Combine the powers of 2. Using $a^m \cdot a^n = a^{m+n}$ on the numerator first, $2^5 \times 2^{-2} = 2^{5+(-2)} = 2^3$ . Then divide using $a^m \div a^n = a^{m-n}$ : $2^3 \div 2^3 = 2^{3-3} = 2^0 = 1$ .

Step 3. Combine the powers of 3. Using $a^m \div a^n = a^{m-n}$ : $3^4 \div 3^6 = 3^{4-6} = 3^{-2}$ .

Step 4. Combine the results and switch the negative exponent to a fraction:

1 \times 3^{-2} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}

The expression simplifies to $\boxed{\dfrac{1}{9}}$ .

Competitive stretch

Push the same expression one step further by evaluating $\left(\dfrac{1}{9}\right)^{-1}$ . A negative exponent on a fraction flips it: $\left(\dfrac{1}{9}\right)^{-1} = 9^1 = 9$ . So while $\dfrac{2^5 \times 2^{-2} \times 3^4}{2^3 \times 3^6} = \dfrac{1}{9}$ , its reciprocal — obtained by nothing more than flipping the sign of the exponent — is exactly $9$ , a fast way to sanity-check an answer without redoing the whole simplification.

Common traps

Treating $a^{-n}$ as a negative number. $2^{-3}$ is $\dfrac{1}{8}$ , a small positive fraction — not $-8$ . The minus sign in the exponent reciprocates; it does not put a minus on the value.
Adding exponents when the bases differ. $a^m \cdot a^n = a^{m+n}$ only works when the base is the same on both sides; $2^3 \times 3^2$ stays as it is and cannot become a single power of $2$ or $3$ .
Applying $a^0 = 1$ when $a = 0$ . $0^0$ is undefined at this level; the rule explicitly requires $a \neq 0$ .
Confusing $(a^m)^n$ with $a^m \times a^n$ . Raising a power to a power multiplies the exponents, giving $a^{mn}$ , not $a^{m+n}$ — this mix-up is one of the most frequent slips in the chapter.

Ask the tutor: Want to see one of the laws proved from scratch with a different base, or work through a trickier mixed-exponent simplification step by step? Ask the AI tutor for extra practice on the laws of exponents.

Why does a negative exponent make a fraction and not a negative number?

Because the exponent only tells you how many times to multiply the base, and a negative sign there is a separate instruction meaning "put it in the denominator instead." Follow the pattern of the division law: $a^0 \div a^n = a^{-n}$ , but $a^0 \div a^n$ is also just $1 \div a^n = \dfrac{1}{a^n}$ by ordinary division. The two must match, so $a^{-n} = \dfrac{1}{a^n}$ — always positive if $a$ is positive, never a sign flip on the value.

Can I use the product law when the bases are different but the exponents are the same, like 2 cubed times 3 cubed?

Yes, but that is a different law from $a^m \cdot a^n = a^{m+n}$ . When the exponents match instead of the bases, use $a^n \times b^n = (ab)^n$ : $2^3 \times 3^3 = (2 \times 3)^3 = 6^3 = 216$ . Check the two numbers side by side before deciding which law applies — same base means add exponents; same exponent means multiply bases first.

Do the laws of exponents work for fractional exponents like a to the power one half?

Not in this chapter. ICSE Class 8 restricts the laws of exponents to integer exponents — positive, negative and zero — applied to a nonzero base. Exponents that are fractions, such as $a^{1/2}$ for a square root, follow the same underlying pattern but are introduced formally only in Class 9, so you will not be asked to simplify them here.