Laws of Exponents for Real Numbers

In earlier classes you used $2^3 = 8$ and $2^4 = 16$. You also met the rule $2^3 \cdot 2^4 = 2^7$. This lesson stretches those rules to real exponents, especially fractions like $2^{1/2}$. Once that step is made, the symbol $\sqrt{2}$ becomes the same thing as $2^{1/2}$ , and the whole machinery of exponents applies to surds for free.

Definitions

For a positive real base $a$ and a positive integer $n$, $a^n = \underbrace{a \cdot a \cdots a}_{n \text{ times}}.$ We extend this to:

• Zero exponent: $a^0 = 1$ (any non-zero base).
• Negative exponent: $a^{-n} = \dfrac{1}{a^n}$.
• Unit fraction exponent: $a^{1/n} = \sqrt[n]{a}$, the positive $n$-th root of $a$ (for $a > 0$).
• Rational exponent: $a^{m/n} = (a^{1/n})^m = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$ (the last equality is a useful identity).

Throughout this lesson $a > 0$ unless otherwise stated; that avoids subtle issues with even roots of negative numbers.

The five laws

For any positive real $a, b$ and any rational numbers $p, q$:

1. Product rule: $a^p \cdot a^q = a^{p+q}$.
2. Quotient rule: $\dfrac{a^p}{a^q} = a^{p-q}$.
3. Power of a power: $(a^p)^q = a^{pq}$.
4. Power of a product: $a^p \cdot b^p = (ab)^p$.
5. Power of a quotient: $\dfrac{a^p}{b^p} = \left(\dfrac{a}{b}\right)^p$.

These are the same five rules you learned for integer exponents , now they hold for all rational exponents as well.

Why the rules extend smoothly. Take rule 1 with $p = \dfrac{1}{2}, q = \dfrac{1}{2}$. The rule predicts $a^{1/2} \cdot a^{1/2} = a^1 = a$. The geometric truth is $\sqrt{a} \cdot \sqrt{a} = a$. The two agree , and demanding all five rules continue to hold is exactly what forces the definition $a^{1/2} = \sqrt{a}$. The definition is not arbitrary; it is the unique extension that keeps the algebra consistent.

A consistency check. Rule 3 with $p = 2, q = \dfrac{1}{2}$ predicts $(a^2)^{1/2} = a^1 = a$. That matches $\sqrt{a^2} = a$ (since $a > 0$). Good. Rule 3 with $p = \dfrac{1}{2}, q = \dfrac{1}{2}$ predicts $(a^{1/2})^{1/2} = a^{1/4}$. That matches $\sqrt[4]{a} = \sqrt{\sqrt{a}}$. Good.

Using the laws on surds. Because $\sqrt{a} = a^{1/2}$, statements about surds reduce to statements about exponents. For instance, $\sqrt{a} \cdot \sqrt{b} = a^{1/2} \cdot b^{1/2} = (ab)^{1/2} = \sqrt{ab}.$ This is the reason for the product rule of surds you used in the last lesson.

Cube roots and higher. $a^{1/3} = \sqrt[3]{a}$ is the unique real $n$-th root of $a$ for $a > 0$. So $8^{1/3} = 2$ and $27^{2/3} = (27^{1/3})^2 = 3^2 = 9$. Negative bases need care: $(-1)^{1/2}$ is not real, but $(-1)^{1/3} = -1$ is.

Real (irrational) exponents. The same rules continue to hold even when $p, q$ are irrational, like $2^\pi$. We will not prove this here, but the idea is that $2^\pi$ is the limit of $2^{r}$ for rational $r$'s approaching $\pi$. This is why expressions like $\pi^e$ make sense.

Worked examples

Example 1. Simplify $2^3 \cdot 2^5$.

Product rule: $2^3 \cdot 2^5 = 2^{3+5} = 2^8 = 256$.

Example 2. Simplify $\dfrac{3^8}{3^5}$.

Quotient rule: $3^{8-5} = 3^3 = 27$.

Example 3. Simplify $5^{1/2} \cdot 5^{1/4}$.

Product rule: $5^{1/2 + 1/4} = 5^{3/4} = (5^3)^{1/4} = 125^{1/4} = \sqrt[4]{125}$.

Example 4. Evaluate $\left(\dfrac{27}{125}\right)^{2/3}$.

Quotient rule and power of a quotient: $= \dfrac{27^{2/3}}{125^{2/3}} = \dfrac{(3^3)^{2/3}}{(5^3)^{2/3}} = \dfrac{3^2}{5^2} = \dfrac{9}{25}$.

Example 5. Simplify $\dfrac{a^{1/2} \cdot a^{2/3}}{a^{1/6}}$.

Product rule on top: $a^{1/2 + 2/3} = a^{7/6}$. Quotient rule: $a^{7/6 - 1/6} = a^{6/6} = a$.

Try it yourself

1. Simplify $2^{1/2} \cdot 2^{3/2}$.
2. Evaluate $16^{3/4}$.
3. Simplify $7^{1/5} \cdot 7^{4/5}$.
4. Simplify $\left(8^{2/3}\right)^{-1/2}$.
5. Simplify $\dfrac{5^{3/2}}{5^{1/2}}$.
6. Evaluate $(64)^{-1/3}$.
7. Show that $\sqrt[6]{27} = \sqrt{3}$ using the laws of exponents.
8. Simplify $\left(\dfrac{16}{81}\right)^{1/4}$.
9. Simplify $\dfrac{a^{1/2}}{a^{-1/3}}$ in the form $a^{\square}$.
10. If $9^x = 27$, find $x$.

Pitfalls / Insight

• Bases must match for product/quotient rules. $2^3 \cdot 3^4$ does not combine into one power.
• Negative exponent is reciprocal, not negative number. $2^{-3} = \dfrac{1}{8}$, not $-8$.
• For real-exponent rules, keep $a > 0$. Things like $(-4)^{1/2}$ are not real numbers and break the laws.

Insight. The phrase "law of exponents" is just rule 1, used five times in slightly different disguises. If you remember $a^p \cdot a^q = a^{p+q}$, you can re-derive every other rule by writing things in terms of products and reciprocals. So practise that one rule until it is automatic , the rest follow.