MCR3U Exponent Laws: Notes and Practice

Every exponent law is really just a shortcut for repeated multiplication. $a^{3}$ means $a \cdot a \cdot a$ — three copies of $a$ multiplied together. Once you think of exponents that way, rules like "add the exponents when multiplying" stop being things to memorize and start being things you can rebuild from scratch if you forget them. Exponents run through this whole unit: an investment at 5% is multiplied by $1.05$ every year, a bacteria count doubles every hour, and scientific notation writes the mass of Earth as $5.97 \times 1 0^{24}$ kg. To handle any of those you need to combine, simplify and evaluate powers quickly and confidently, often without a calculator.

Fig 1Each exponent law expands directly from counting repeated factors of the base.

The core laws

Write the factors out and each law explains itself. $a^{3} \cdot a^{2} = (a \cdot a \cdot a) (a \cdot a) = a^{5}$ : three factors plus two factors is five factors, so the exponents add. In $\frac{a ^{5}}{a ^{2}} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a}$ , two factors of $a$ cancel from the top and bottom, leaving $a^{3}$ , so the exponents subtract. And $(a^{2})^{3} = a^{2} \cdot a^{2} \cdot a^{2} = a^{2 + 2 + 2} = a^{6} :$ three groups of two factors is six factors, so the exponents multiply. Every row of the table below is one of these counting arguments.

Law	Rule	Why
Product	$a^{m} \cdot a^{n} = a^{m + n}$	Combining two groups of repeated factors adds the counts.
Quotient	$\frac{a ^{m}}{a ^{n}} = a^{m - n}$	Cancelling common factors from top and bottom subtracts the counts.
Power of a power	$(a^{m})^{n} = a^{mn}$	n copies of m factors each is m×n factors total.
Power of a product	$(ab)^{n} = a^{n} b^{n}$	Each factor in the product gets raised to the power separately.
Power of a quotient	$(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$	Numerator and denominator are each raised to the power.

Same base, and products only

The laws combine powers of the same base:

2^{3} \cdot 2^{4} = 2^{7}

, but

2^{3} \cdot 3^{4}

cannot be merged into one power. They also work only across multiplication and division, never addition:

(a + b)^{2} \neq = a^{2} + b^{2}

(try

a = b = 1

4 \neq = 2

), and

2^{3} + 2^{4} = 8 + 16 = 24

, not

2^{7}

. When the bases differ but are related, such as 4 and 8, rewrite both as powers of 2 first:

4^{3} \cdot 8 = (2^{2})^{3} \cdot 2^{3} = 2^{6} \cdot 2^{3} = 2^{9} .

Zero and negative exponents

Zero and negative exponents are defined so the pattern of the quotient law keeps working even when the top exponent is smaller than the bottom. Since $\frac{a ^{n}}{a ^{n}} = 1$ but also $\frac{a ^{n}}{a ^{n}} = a^{n - n} = a^{0}$ by the quotient law, we must have $a^{0} = 1$ (for $a \neq = 0$ ). Similarly, $\frac{a ^{2}}{a ^{5}} = \frac{1}{a ^{3}}$ by cancelling, and $a^{2 - 5} = a^{- 3}$ by the quotient law, so $a^{- 3} = \frac{1}{a ^{3}}$ . A negative exponent flips the base to the other side of a fraction; it does not make the value negative. The base $0$ is left out on purpose: $0^{0}$ is left undefined in this course (the patterns $0^{n} = 0$ and $a^{0} = 1$ disagree about it) and $0^{- 1}$ would mean $\frac{1}{0}$ .

Power of 2	Value	How it got there
$2^{3}$	8	start
$2^{2}$	4	divide by 2
$2^{1}$	2	divide by 2
$2^{0}$	1	divide by 2
$2^{- 1}$	$\frac{1}{2}$	divide by 2
$2^{- 2}$	$\frac{1}{4}$	divide by 2

Lowering the exponent by 1 divides by the base, so the pattern itself forces 2 to the 0 to equal 1 and 2 to the -1 to equal one half.

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

VideoA quick run through each exponent rule with examples, from the product and quotient laws through zero and negative exponents.Watch on YouTube

Rational exponents

A rational (fractional) exponent combines a root and a power. Why should $9^{1/2}$ mean $9$ ? Apply the power-of-a-power law: $(9^{1/2})^{2} = 9^{\frac{1}{2} \cdot 2} = 9^{1} = 9$ . So $9^{1/2}$ is a positive number whose square is 9, which is exactly $9 = 3$ . In general $(a^{1/ n})^{n} = a$ , so $a^{1/ n} = n a$ , and then $a^{m / n} = (a^{1/ n})^{m}$ . The denominator of the fraction is the root, and the numerator is the power. This lets you rewrite any radical as a power, such as $3 x^{2} = x^{2/3}$ and $\frac{1}{x} = x^{- 1/2}$ , and then use the ordinary laws on it.

a^{\frac{m}{n}} = (n a)^{m} = n a^{m}

Which order is easier?

Taking the root first, then the power, usually keeps the numbers smaller and easier to compute by hand — e.g. for

8^{2/3}

, compute

38 = 2

first, then square it, instead of computing

8^{2} = 64

first and taking

364

. With a big base the difference is dramatic:

6 4^{5/6}

(664)^{5} = 2^{5} = 32

, while the other order needs the sixth root of

6 4^{5} = 1073741824

. Rational exponents also let you combine radicals with different indices, which is impossible in radical form:

x \cdot 3 x = x^{1/2} \cdot x^{1/3} = x^{3/6 + 2/6} = x^{5/6} .

VideoAn MCR3U overview of rational exponents: the denominator of the exponent is the root and the numerator is the power.Watch on YouTube

A reliable simplifying routine

Deal with brackets first, applying the outer exponent to every factor inside, coefficients included: $(2 x^{3})^{4} = 2^{4} x^{12} = 16 x^{12}$ , not $2 x^{12}$ .
Combine powers of the same base: add exponents when multiplying, subtract when dividing (top exponent minus bottom exponent).
Evaluate numerical powers, taking the root before the power for rational exponents: $2 5^{3/2} = 5^{3} = 125$ .
Rewrite any negative exponents as reciprocals at the very end, so the final answer has positive exponents only.
Check signs: $(- 8)^{1/3} = - 2$ is fine (odd root), $(- 16)^{1/4}$ is not a real number (even root of a negative), and $- 1 6^{1/4} = - (1 6^{1/4}) = - 2$ , because the exponent acts before the minus sign.

Exponent Laws

Key things to remember

The core laws

Zero and negative exponents

Rational exponents

A reliable simplifying routine

Worked examples

Where marks get lost

Flashcards 9

Practice