1.4.1 Exponents and their rules

What an exponent means

An exponent counts how many copies of a number are multiplied together. In $2^5$ the base is 2 and the exponent is 5:

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

Read $2^5$ as "2 to the power 5", or "2 to the fifth". A power of 1 is the number itself: $7^1 = 7$ .

The exponent is a count, so it has no units. If the base has units, they are raised too: a square with sides of 3 cm has an area of $(3\text{ cm})^2 = 9\text{ cm}^2$ .

1.5 \times 1.5 \times 1.5

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Multiplying and dividing powers

Write the factors out and count them:

2^3 \times 2^2 = (2 \times 2 \times 2) \times (2 \times 2) = 2^5

Three factors and two more make five. The same counting works for any base.

Rule: Product rule

a^m \cdot a^n = a^{m+n}

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

Dividing cancels factors instead. Two of the threes on top cancel the two below:

\frac{3^5}{3^2} = \frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3} = 3^3

Rule: Quotient rule

\dfrac{a^m}{a^n} = a^{m-n}

When you divide two powers of the same base, subtract the bottom exponent from the top one.

Try it Check both rules with numbers. Each line prints two values that should match. Division in Python always gives a decimal, which is why one value shows as 27.0 .

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Powers of powers, products and quotients

Squaring $2^3$ means writing it twice. Then the product rule adds the exponents:

(2^3)^2 = 2^3 \times 2^3 = 2^{3+3} = 2^6

Two copies of three factors is six factors, so the exponents multiply.

Rule: Power of a power

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

Raising a power to another power multiplies the exponents.

Rule: Power of a product or quotient

\begin{gathered} (ab)^n = a^n b^n \\ \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \end{gathered}

A power outside brackets applies to every factor inside them. Read backwards, it joins different bases that share an exponent: $2^3 \times 3^3 = 6^3$ .

(2 \times 5)^3 = 10^3

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A worked simplification

Most simplifications use several rules in a row. Here is $\dfrac{(2x^3)^2 \cdot x^4}{4x^5}$ , one rule at a time:

\begin{aligned} \frac{(2x^3)^2 \cdot x^4}{4x^5} &= \frac{4x^6 \cdot x^4}{4x^5} \\[4pt] &= \frac{4x^{10}}{4x^5} \\[4pt] &= x^5 \end{aligned}

The first line squares each factor in the bracket: $2^2 = 4$ , and $(x^3)^2 = x^6$ by the power of a power. The second adds exponents by the product rule. The third cancels the 4s and subtracts exponents by the quotient rule.

x

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Zero and negative exponents

Watch what happens as the exponent goes down by one. Each step divides by 2:

$2^3$	$2^2$	$2^1$	$2^0$	$2^{-1}$	$2^{-2}$
8	4	2	1	$\tfrac{1}{2}$	$\tfrac{1}{4}$

Keep the pattern going past $2^1$ and you reach $2^0 = 1$ , then fractions. They get smaller at every step, but halving never reaches 0.

The powers of 2 from $2^{-3}$ to $2^4$ . Each step up in the exponent doubles the bar; each step down halves it, so negative exponents give values between 0 and 1.

The quotient rule gives the same answers. $\frac{2^3}{2^3}$ is 1, and the rule says it is $2^{3-3} = 2^0$ . Likewise $\frac{2^2}{2^5} = \frac{4}{32} = \frac{1}{8}$ , and the rule says it is $2^{2-5} = 2^{-3}$ .

Rule: Zero and negative exponents

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

Any nonzero number to the power 0 is 1. A negative exponent means "one over": it does not make the number negative.

So $5^{-1} = \frac{1}{5} = 0.2$ , the reciprocal. A negative power of a fraction flips it: $\left(\tfrac{2}{3}\right)^{-2} = \left(\tfrac{3}{2}\right)^2 = \tfrac{9}{4}$ .

Try it The loop counts the exponent down from 3 to -3 and prints each power of 2. Notice every power is positive, even when the exponent is negative.

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A base of zero

$0^n = 0$ for any positive whole number $n$ , because a product of zeros is zero. A negative power of 0 would mean $\frac{1}{0}$ , which does not exist. So $0^{-1}$ has no value.

$0^0$ is the one clash. The powers $0^1, 0^2, 0^3$ suggest 0, while $a^0 = 1$ suggests 1. Algebra and Python take $0^0 = 1$ , but in calculus a limit of that shape is not always 1.

Try it Python agrees with each case. 0 ** -1 stops with a ZeroDivisionError, which the cell catches and reports.

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Negative bases

A negative base is fine with a whole-number exponent. Each extra factor of $-2$ flips the sign:

$(-2)^1$	$(-2)^2$	$(-2)^3$	$(-2)^4$	$(-2)^5$
$-2$	4	$-8$	16	$-32$

An even exponent gives a positive answer; an odd one keeps the minus sign. The brackets matter. $(-2)^2 = 4$ , but $-2^2$ means $-(2^2) = -4$ , because powers come before the minus sign (lesson 0.3.1).

Try it Python follows the same order: -2**2 squares first, then applies the minus.

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Fractional exponents

What should $9^{1/2}$ mean? Square it and use the power-of-a-power rule:

\left(9^{1/2}\right)^2 = 9^{\frac{1}{2} \times 2} = 9^1 = 9

So $9^{1/2}$ is the number whose square is 9. That is 3, the square root of 9.

Rule: Fractional exponents

\begin{gathered} a^{1/n} = \sqrt[n]{a} \\ a^{m/n} = \left(\sqrt[n]{a}\right)^m \\ \text{for } a > 0 \end{gathered}

The bottom of the fraction is a root. The top is an ordinary power. Take the root first and the numbers stay small: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ .

A negative fractional exponent combines the two ideas: root, power, then one over. So $8^{-2/3} = \frac{1}{8^{2/3}} = \frac{1}{4}$ .

Any decimal exponent works for a positive base: $10^{0.5} = \sqrt{10} \approx 3.1623$ . Lesson 1.4.2 uses this to let the exponent be any number at all.

\tfrac{2}{3}

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Why the base must be positive

No real number squares to $-4$ , so $(-4)^{1/2}$ has no real value. That is why lesson 1.1.1 left negative inputs out of the domain of $\sqrt{x}$ . Every even root of a negative number fails the same way.

Odd roots of negatives do exist: $(-2)^3 = -8$ , so the cube root of $-8$ is $-2$ . But $(-8)^{2/3}$ is unsafe, because $\frac{2}{3}$ is also $\frac{4}{6}$ , which asks for an even root. So the rule keeps to $a > 0$ .

Try it Python shows the trap. (-8) ** (1/3) does not print -2. It switches to complex numbers (lesson 1.7.2), marked by the j, and returns a different cube root. For the real cube root, use np.cbrt .

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Common mistakes

(3 + 4)^2

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Exponent rules

0 of 12 correct

1.
Simplify $x^3 \cdot x^4$
2.
Simplify $\dfrac{y^9}{y^4}$
3.
Simplify $(a^2)^5$
4.
Simplify $\dfrac{(3x^2)^2 \cdot x}{9x^3}$
5.
Evaluate $5^0$
6.
Evaluate $2^{-3}$ (a fraction or a decimal)
7.
Evaluate $\left(\tfrac{2}{5}\right)^{-2}$
8.
Evaluate $(-3)^3$
9.
Evaluate $-3^2$
10.
Evaluate $27^{2/3}$
11.
Evaluate $16^{-1/2}$
12.
Rewrite $x^{-2}$ without a negative exponent

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

a^n

Exponents and their rules

What an exponent means

Multiplying and dividing powers

Rule: Product rule

Rule: Quotient rule

Powers of powers, products and quotients

Rule: Power of a power

Rule: Power of a product or quotient

A worked simplification

Zero and negative exponents

Rule: Zero and negative exponents

A base of zero

Negative bases

Fractional exponents

Rule: Fractional exponents

Why the base must be positive

Common mistakes

Exponent rules

Summary card

Free further reading