What is the difference between a to the power -n and -a to the power n?

a to the power -n is the reciprocal 1 over a to the power n, while -a to the power n is the negative of a to the power n. For example, 2 to the power -3 = 1 over 8 but -2 to the power 3 = -8.

Should I take the root or the power first in a to the power m over n?

Both give the same answer for a positive base, but taking the root first keeps the numbers small: 8 to the power 2 over 3 = (the cube root of 8) to the power 2 = 4 is easier than the cube root of 64 = 4.

Laws of Exponents: Rules, Formulas and Examples

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

Because the quotient rule must keep working. a to the power n over a to the power n is 1 for any non-zero a, and the quotient rule says it is also a to the power n-n = a to the power 0. So a to the power 0 has to be 1.

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

Tiles showing three 2s times four 2s making seven 2s, so 2^3 x 2^4 = 2^7 = 128; and six 5s over two 5s with two pairs struck out, leaving 5^4 = 625. — Counting factors: multiplying adds them, dividing cancels them in pairs.

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

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

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, $\displaystyle \frac{5^2}{5^2} = 5^{2-2} = 5^0$ . 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: $\displaystyle \frac{a^0}{a^n} = a^{0-n} = a^{-n}$ , and $\displaystyle \frac{a^0}{a^n} = \frac{1}{a^n}$ .

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

Graph of y = 2^x for x from -3 to 3 with points marked: 2^-3 = 1/8, 2^-2 = 1/4, 2^-1 = 1/2, 2^0 = 1, 2^1 = 2, 2^2 = 4 and 2^3 = 8. — The graph of $y = 2^x$ : moving one step left halves the value, which is why $2^0 = 1$ and $\displaystyle 2^{-1} = \frac{1}{2}$ .

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

Write every number as a power of a prime (for example $8 = 2^3$ , $27 = 3^3$ , $81 = 3^4$ ).
Remove brackets using the power of a product, power of a quotient and power of a power rules.
Collect powers of the same base using the product and quotient rules.
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 $\displaystyle \frac{(x^3y^2)^4}{x^5y^3}$ .

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

Example 3: a negative fractional exponent

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

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

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

Rule	Formula	Example
Product	$a^m \times a^n = a^{m+n}$	$x^2 \times x^3 = x^5$
Quotient	$a^m \div a^n = a^{m-n}$	$y^5 \div y^2 = y^3$
Power of a power	$(a^m)^n = a^{mn}$	$(2^3)^2 = 2^6 = 64$
Power of a product	$(ab)^n = a^nb^n$	$(2x)^3 = 8x^3$
Power of a quotient	$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\displaystyle \left(\frac{2}{3}\right)^4 = \frac{16}{81}$
Zero power	$a^0 = 1$	$1{,}000{,}000^0 = 1$
Negative power	$\displaystyle a^{-n} = \frac{1}{a^n}$	$\displaystyle 3^{-2} = \frac{1}{9}$
Fractional power	$\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}$	$\displaystyle 16^{\frac{1}{4}} = 2$

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

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

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

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Laws of Exponents: Exponent Rules and Formulas with Examples

What exactly is an expo­nent?#

Base and expo­nent#

Read­ing pow­ers aloud#

The core expo­nent laws#

1. The prod­uct rule#

2. The quo­tient rule#

3. The power of a power rule#

Han­dling dif­fer­ent bases and brack­ets#

4. The power of a prod­uct rule#

5. The power of a quo­tient rule#

The spe­cial cases: zero and neg­a­tive expo­nents#

6. The zero expo­nent rule#

7. The neg­a­tive expo­nent rule#

Frac­tional (ratio­nal) expo­nents#

8. The ratio­nal expo­nent rule#

A method for sim­pli­fy­ing any expres­sion#

Worked exam­ples#

Exam­ple 1: com­bin­ing rules with one base#

Exam­ple 2: alge­braic bases#

Exam­ple 3: a neg­a­tive frac­tional expo­nent#

Exam­ple 4: neg­a­tive expo­nents inside a sum#

Exam­ple 5: mixed bases#

Exam­ple 6: solv­ing for an unknown expo­nent#

Sum­mary table for quick ref­er­ence#

Com­mon mis­takes#

Try these#

Key terms#

Com­mon ques­tions#

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

Can I add expo­nents when the bases are dif­fer­ent?#

What is the dif­fer­ence between a−na^{-n}a−n and −an-a^n−an?#

Should I take the root or the power first in amn\displaystyle a^{\frac{m}{n}}anm​?#

Where are expo­nents used in Class 10?#

Ref­er­ences#

Related read­ing#