Laws of Exponents: Exponent Rules and Formulas with Examples

An expo­nent is a short way of writ­ing repeated mul­ti­pli­ca­tion. Instead of $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ we write $2^{10}$, which equals 1024. Expo­nents appear wher­ever quan­ti­ties grow or shrink by the same fac­tor again and again: com­pound inter­est, pop­u­la­tion growth, sci­en­tific nota­tion for very large and very small num­bers, and the prime fac­tori­sa­tions you will use in Class 10. Behind the large num­bers sits a small set of rules. Once you know them, and why they work, you can sim­plify long expres­sions in a few lines.

What exactly is an expo­nent?

Base and expo­nent

An expo­nent tells you how many times to mul­ti­ply a num­ber, the base, by itself. In the expres­sion $a^n$:

• $a$ is the base;
• $n$ is the expo­nent (also called the power or index).

For a pos­i­tive inte­ger $n$,

$$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ factors}}$$

So $3^4$ is not $3 \times 4$. It is $3 \times 3 \times 3 \times 3$, which equals $81$.

Read­ing pow­ers aloud

$a^2$ is read "$a$ squared", $a^3$ is "$a$ cubed", and $a^n$ is "$a$ to the power $n$". Note that $-3^2 = -(3^2) = -9$, while $(-3)^2 = 9$: the expo­nent applies only to what sits directly below it.

The core expo­nent laws

The laws below hold for any non-zero bases $a$ and $b$. We first see why each is true for pos­i­tive whole-num­ber expo­nents, where we can sim­ply count fac­tors. The def­i­n­i­tions of zero, neg­a­tive and frac­tional expo­nents later in the les­son are cho­sen so that the same laws keep work­ing.

1. The prod­uct rule

When you mul­ti­ply two pow­ers with the same base, add the expo­nents.

$$a^m \times a^n = a^{m+n}$$

Why it works: $a^m$ con­tributes $m$ fac­tors of $a$ and $a^n$ con­tributes $n$ more, so the prod­uct has $m + n$ fac­tors of $a$.

Exam­ple: $2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7 = 128$.

2. The quo­tient rule

When you divide two pow­ers with the same base, sub­tract the expo­nent of the denom­i­na­tor from the expo­nent of the numer­a­tor.

$$\displaystyle \frac{a^m}{a^n} = a^{m-n}$$

Why it works: each fac­tor $a$ in the denom­i­na­tor can­cels one fac­tor $a$ in the numer­a­tor, so $n$ of the $m$ fac­tors dis­ap­pear.

Exam­ple: $\displaystyle \dfrac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$.

3. The power of a power rule

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

$$(a^m)^n = a^{mn}$$

Why it works: $(a^m)^n$ is $n$ copies of $a^m$ mul­ti­plied together, and each copy has $m$ fac­tors, giv­ing $m \times n$ fac­tors in all.

Exam­ple: $(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$.

Han­dling dif­fer­ent bases and brack­ets

Some­times the base is a prod­uct or a quo­tient. The power then applies to each part.

4. The power of a prod­uct rule

$$(ab)^n = a^n \times b^n$$

Why it works: $(ab)^n = (ab)(ab)\cdots(ab)$ with $n$ brack­ets. Rear­rang­ing gives $n$ fac­tors of $a$ and $n$ fac­tors of $b$.

Exam­ple: $(2x)^3 = 2^3 \times x^3 = 8x^3$.

5. The power of a quo­tient rule

$$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Exam­ple: $\displaystyle \left(\dfrac{2}{3}\right)^4 = \dfrac{2^4}{3^4} = \dfrac{16}{81}$.

The spe­cial cases: zero and neg­a­tive expo­nents

These are the rules that most often cause mis­takes in exams, yet they fol­low directly from the quo­tient rule.

6. The zero expo­nent rule

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

$$a^0 = 1 \quad (a \neq 0)$$

Why: by the quo­tient rule, $\displaystyle \dfrac{5^2}{5^2} = 5^{2-2} = 5^0$. But any non-zero num­ber divided by itself is 1, so $5^0$ must be 1. The expres­sion $0^0$ is left unde­fined in school math­e­mat­ics.

7. The neg­a­tive expo­nent rule

A neg­a­tive expo­nent means "take the rec­i­p­ro­cal".

$$\displaystyle a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$$

Why: $\displaystyle \dfrac{a^0}{a^n} = a^{0-n} = a^{-n}$, and $\displaystyle \dfrac{a^0}{a^n} = \dfrac{1}{a^n}$.

Exam­ple: $\displaystyle 4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}$. A use­ful con­se­quence is $\displaystyle \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}$.

Frac­tional (ratio­nal) expo­nents

8. The ratio­nal expo­nent rule

When the expo­nent is a frac­tion, it describes a root. The denom­i­na­tor of the frac­tion is the index of the root. For a pos­i­tive base $a$,

$$\displaystyle a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$

Why: by the power of a power rule, $\displaystyle \left(a^{\frac{1}{n}}\right)^n = a^{\frac{n}{n}} = a^1 = a$. So $\displaystyle a^{\frac{1}{n}}$ is the num­ber whose $n$th power is $a$, which is exactly $\sqrt[n]{a}$.

Exam­ple: $\displaystyle 9^{\frac{1}{2}} = \sqrt{9} = 3$.

Exam­ple: $\displaystyle 8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$.

Tak­ing the root first, as here, keeps the num­bers small.

A method for sim­pli­fy­ing any expres­sion

1. Write every num­ber as a power of a prime (for exam­ple $8 = 2^3$, $27 = 3^3$, $81 = 3^4$).
2. Remove brack­ets using the power of a prod­uct, power of a quo­tient and power of a power rules.
3. Col­lect pow­ers of the same base using the prod­uct and quo­tient rules.
4. Turn any neg­a­tive expo­nents into rec­i­p­ro­cals at the end, and eval­u­ate.

Worked exam­ples

Exam­ple 1: com­bin­ing rules with one base

Sim­plify $2^5 \times 2^{-3} \div 2^{-1}$.

$$\begin{aligned} 2^5 \times 2^{-3} \div 2^{-1} &= 2^{5 + (-3) - (-1)} \\ &= 2^{3} \\ &= 8 \end{aligned}$$

Exam­ple 2: alge­braic bases

Sim­plify $\displaystyle \dfrac{(x^3y^2)^4}{x^5y^3}$.

$$\displaystyle \begin{aligned} \frac{(x^3y^2)^4}{x^5y^3} &= \frac{x^{12}y^{8}}{x^5y^3} \\ &= x^{12-5}\,y^{8-3} \\ &= x^7y^5 \end{aligned}$$

Exam­ple 3: a neg­a­tive frac­tional expo­nent

Eval­u­ate $\displaystyle \left(\dfrac{27}{8}\right)^{-\frac{2}{3}}$.

$$\displaystyle \begin{aligned} \left(\frac{27}{8}\right)^{-\frac{2}{3}} &= \left(\frac{8}{27}\right)^{\frac{2}{3}} = \left(\frac{2^3}{3^3}\right)^{\frac{2}{3}} \\ &= \frac{2^{2}}{3^{2}} = \frac{4}{9} \end{aligned}$$

Exam­ple 4: neg­a­tive expo­nents inside a sum

Eval­u­ate $\left(3^{-1} + 2^{-1}\right)^{-1}$. The laws do not apply to a sum, so work out the bracket first.

$$\displaystyle \begin{aligned} \left(3^{-1} + 2^{-1}\right)^{-1} &= \left(\frac{1}{3} + \frac{1}{2}\right)^{-1} = \left(\frac{5}{6}\right)^{-1} \\ &= \frac{6}{5} \end{aligned}$$

Exam­ple 5: mixed bases

Eval­u­ate $\displaystyle 25^{\frac{3}{2}} \times 81^{-\frac{1}{4}}$.

$$\displaystyle \begin{aligned} 25^{\frac{3}{2}} \times 81^{-\frac{1}{4}} &= (5^2)^{\frac{3}{2}} \times (3^4)^{-\frac{1}{4}} \\ &= 5^3 \times 3^{-1} \\ &= \frac{125}{3} \end{aligned}$$

Exam­ple 6: solv­ing for an unknown expo­nent

Solve $2^{x+1} = 32$.

Write 32 as a power of 2: $32 = 2^5$. If two pow­ers of the same base (other than 0, 1 or $-1$) are equal, their expo­nents are equal.

$$\begin{aligned} 2^{x+1} &= 2^5 \\ x + 1 &= 5 \\ x &= 4 \end{aligned}$$

Sum­mary table for quick ref­er­ence

Com­mon mis­takes

• Using the laws when the bases are dif­fer­ent. $2^3 \times 3^2$ is not $6^5$; work each power out sep­a­rately: $8 \times 9 = 72$.
• Mul­ti­ply­ing the base by the expo­nent: $3^4$ is $81$, not $12$.
• Adding expo­nents in a sum: $2^3 + 2^4 = 8 + 16 = 24$, which is not $2^7$. The prod­uct rule is for prod­ucts only.
• Think­ing a neg­a­tive expo­nent makes a num­ber neg­a­tive: $\displaystyle 4^{-2} = \frac{1}{16}$, which is pos­i­tive.
• Writ­ing $a^0 = 0$. For any non-zero $a$, $a^0 = 1$.
• For­get­ting brack­ets: $-3^2 = -9$ but $(-3)^2 = 9$, and $2x^3$ means $2 \times x^3$, not $(2x)^3$.

Try these

1. Sim­plify $5^3 \times 5^{-5}$. Answer: $\displaystyle \frac{1}{25}$
2. Sim­plify $(a^2b^{-1})^3$. Answer: $\displaystyle \frac{a^6}{b^3}$
3. Eval­u­ate $\displaystyle 16^{\frac{3}{4}}$. Answer: $8$
4. Eval­u­ate $\displaystyle \left(\frac{1}{2}\right)^{-3}$. Answer: $8$
5. Solve $3^{2x-1} = 27$. Answer: $x = 2$
6. Eval­u­ate $\displaystyle 32^{-\frac{2}{5}}$. Answer: $\displaystyle \frac{1}{4}$

Key terms

Base

The num­ber or expres­sion being mul­ti­plied repeat­edly; in $a^n$ it is $a$.

Expo­nent (power, index)

The num­ber that says how many times the base is used as a fac­tor; in $a^n$ it is $n$.

Power

The whole expres­sion $a^n$, or its value.

Rec­i­p­ro­cal

The num­ber that mul­ti­plies with a given non-zero num­ber to give 1; the rec­i­p­ro­cal of $a^n$ is $a^{-n}$.

Ratio­nal expo­nent

An expo­nent that is a frac­tion, such as $\displaystyle \frac{2}{3}$; it com­bines a power and a root.

$n$th root

For a pos­i­tive num­ber $a$, the pos­i­tive num­ber whose $n$th power is $a$, writ­ten $\sqrt[n]{a}$ or $\displaystyle a^{\frac{1}{n}}$.

Expo­nen­tial growth

Growth by the same fac­tor in each step, as in $1, 2, 4, 8, \ldots$, described by pow­ers such as $2^x$.

Com­mon ques­tions

Why is any num­ber to the power zero equal to 1?

Because the quo­tient rule must keep work­ing. $\displaystyle \frac{a^n}{a^n}$ is 1 for any non-zero $a$, and the quo­tient rule says it is also $a^{n-n} = a^0$. So $a^0$ has to be 1.

Can I add expo­nents when the bases are dif­fer­ent?

No. The prod­uct and quo­tient rules need the same base. If the expo­nents are equal you can com­bine the bases instead, as in $2^3 \times 5^3 = (2 \times 5)^3 = 1000$.

What is the dif­fer­ence between $a^{-n}$ and $-a^n$?

$a^{-n}$ is the rec­i­p­ro­cal $\displaystyle \frac{1}{a^n}$, while $-a^n$ is the neg­a­tive of $a^n$. For exam­ple, $\displaystyle 2^{-3} = \frac{1}{8}$ but $-2^3 = -8$.

Should I take the root or the power first in $\displaystyle a^{\frac{m}{n}}$?

Both give the same answer for a pos­i­tive base, but tak­ing the root first keeps the num­bers small: $\displaystyle 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 4$ is eas­ier than $\sqrt[3]{64} = 4$.

Where are expo­nents used in Class 10?

In the Real Num­bers chap­ter, every num­ber is writ­ten as a prod­uct of prime pow­ers, and the HCF and LCM are found by com­par­ing expo­nents. Expo­nents also appear in sci­en­tific nota­tion and in com­pound inter­est.

Ref­er­ences

1. National Coun­cil of Edu­ca­tional Research and Train­ing. Math­e­mat­ics: Text­book for Class X. NCERT, New Delhi.
2. Gelfand, I. M. and Shen, A. Alge­bra. Birkhäuser.
3. Stew­art, J., Redlin, L. and Wat­son, S. Pre­cal­cu­lus: Math­e­mat­ics for Cal­cu­lus. Cen­gage Learn­ing.

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