Rational exponents (also called fractional indices) let you write radicals using exponent notation and then use the familiar laws of exponents to simplify expressions.
Rational Exponents Are Another Way to Write Roots
- A unit fraction is a fraction with numerator 1 (for example, $\frac{1}{2}$ or $\frac{1}{7}$).
- Unit fractions appear naturally when we write roots as exponents.
Rational exponent
An exponent of the form $\frac{m}{n}$ where $m$ and $n$ are integers and $n\neq 0$.
- For any positive real number $x$, and integers $m$ and $n$ with $n\neq 0$: $$x^{\frac{m}{n}}=(\sqrt[n]{x})^m=\sqrt[n]{x^m}$$
- This statement contains three equivalent ways to see the same number or expression:
- $x^{\frac{m}{n}}$ is exponent form.
- $(\sqrt[n]{x})^m$ means "take the $n$th root, then raise to the power $m$".
- $\sqrt[n]{x^m}$ means "raise to the power $m$, then take the $n$th root".
- The restriction "$x$ is a positive real number" is not just a detail.
- It prevents ambiguity and non-real results when even roots of negative numbers appear (explained later).
The Key Link: Roots Correspond to Unit Fractions
- The most basic rational exponents are unit fractions: $$x^{\frac{1}{n}}=\sqrt[n]{x}$$
- Then you can build other rational exponents by applying a further integer power: $$x^{\frac{m}{n}}=\left(x^{\frac{1}{n}}\right)^m$$
Think of $x^{\frac{m}{n}}$ as a two-step machine: "root" is set by the denominator $n$, and "power" is set by the numerator $m$.
A Reliable Method to Evaluate Expressions Like $a^{m/n}$
To evaluate a number with a rational exponent, it helps to separate the exponent into a unit fraction and an integer.
Simplify $16^{\frac{3}{4}}$ without a calculator.
Solution
Split the exponent: $$16^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^3$$
Now evaluate the fourth root: $$16^{\frac{1}{4}}=\sqrt[4]{16}=2$$
So: $$16^{\frac{3}{4}}=2^3=8$$
You could also do it in the other order: $$16^{\frac{3}{4}}=\left(16^3\right)^{\frac{1}{4}}=\sqrt[4]{4096}=8$$
- When the base is a perfect power (like $16=2^4$), "take the root first" is usually faster.
- When the exponent makes the power convenient first, "power first" can be easier.
- Both are valid as long as you stay consistent.
Negative Rational Exponents Mean 'Reciprocal' First
- A negative exponent means take the reciprocal: $$x^{-k}=\frac{1}{x^k} \quad (x\neq 0)$$
- The same idea works for rational exponents: $$x^{-\frac{m}{n}}=\frac{1}{x^{\frac{m}{n}}}$$
Simplify $\left(\frac{27}{8}\right)^{-\frac{2}{3}}$.
Solution
- Deal with the negative exponent by taking the reciprocal: $$\left(\frac{27}{8}\right)^{-\frac{2}{3}}=\left(\frac{8}{27}\right)^{\frac{2}{3}}$$
- Now split $\frac{2}{3}$ into "cube root then square": $$\left(\frac{8}{27}\right)^{\frac{2}{3}}=\left(\left(\frac{8}{27}\right)^{\frac{1}{3}}\right)^2$$
- Cube root: $$\left(\frac{8}{27}\right)^{\frac{1}{3}}=\frac{2}{3}$$
- Square: $$\left(\frac{2}{3}\right)^2=\frac{4}{9}$$
- Don't forget that the negative sign applies to the whole exponent.
- For example, $x^{-2/3}$ is not $-x^{2/3}$, it is $\frac{1}{x^{2/3}}$.
The Laws of Exponents Still Work with Rational Exponents
Once expressions are written with exponents, you can simplify using the standard index laws (for appropriate values of the base):
- Product rule: $x^a\,x^b=x^{a+b}$
- Quotient rule: $\frac{x^a}{x^b}=x^{a-b}$
- Power of a power: $(x^a)^b=x^{ab}$
- These rules are especially powerful for radicals because radicals are really rational exponents.
- For a recap, you can go through the previous article about exponent laws.
Multiplying and Dividing Radicals with the Same Radicand
When the radicand (the number under the root sign) is the same but the indices differ, write as powers and add or subtract exponents.
For the same base $a$: $$\sqrt[n]{a}\,\sqrt[m]{a}=a^{\frac{1}{n}}\,a^{\frac{1}{m}}=a^{\frac{1}{n}+\frac{1}{m}}=a^{\frac{n+m}{nm}}$$
$$\text{and}$$
$$\frac{\sqrt[n]{a}}{\sqrt[m]{a}}=\frac{a^{\frac{1}{n}}}{a^{\frac{1}{m}}}=a^{\frac{1}{n}-\frac{1}{m}}=a^{\frac{n-m}{nm}}$$
Simplify $\frac{\sqrt[4]{8}}{\sqrt{2}}$.
Solution
- Rewrite with a common base (prime powers help): $$\frac{\sqrt[4]{8}}{\sqrt{2}}=\frac{\sqrt[4]{2^3}}{\sqrt{2}}=\frac{2^{\frac{3}{4}}}{2^{\frac{1}{2}}}$$
- Subtract exponents: $$2^{\frac{3}{4}-\frac{1}{2}}=2^{\frac{3}{4}-\frac{2}{4}}=2^{\frac{1}{4}}=\sqrt[4]{2}$$
- If you see different roots of the same base (like $\sqrt[4]{2}$ and $\sqrt{2}$), convert to powers of the same base immediately.
- It prevents messy radical manipulation.
When Radicals Cannot Be Simplified by Exponent Rules
- The exponent laws combine terms only when the bases match.
- That is why expressions like these do not simplify using index laws:
- $\sqrt[3]{4}\,\sqrt[5]{3}$ (different bases, no relationship)
- $\sqrt[4]{144}\,\sqrt{125}$ (different radicands, not powers of one common base)
- $\frac{\sqrt{5}}{\sqrt[3]{2}}$ (different bases)
- You can simplify when the radicands can be rewritten as powers of a common base (often using prime factorization).
- A good habit is to rewrite radicands as prime powers (like $8=2^3$, $75=3\cdot 5^2$).
- Then you can clearly see when bases match.
Why the Base Is Usually Restricted to Positive Real Numbers
Rational exponents are defined (in this course) for positive real bases because of how roots behave in the real numbers.
- Consider $\sqrt{-8}$.
- Writing it as a power gives $(-8)^{\frac{1}{2}}$.
- But the square root of a negative is not a real number, so $(-8)^{\frac{1}{2}}$ is not real.
- However, odd roots of negative numbers do exist in the real numbers: $$\sqrt[3]{-8}=-2 \quad \Rightarrow \quad (-8)^{\frac{1}{3}}=-2$$
The problem appears when the same rational number can be written in different fractional forms.
- For example: $$\frac{1}{3}=\frac{2}{6}$$
- So if you allow negative bases freely, you might try to say: $$(-8)^{\frac{1}{3}} = (-8)^{\frac{2}{6}}$$
- But $(-8)^{\frac{2}{6}}=\left((-8)^{\frac{1}{6}}\right)^2$ involves a 6th root of a negative number, which is not real.
- This is why, to keep the definition consistent and always real-valued, we insist on $x>0$ when defining $x^{m/n}$ in the real number system.
- Do not assume $a^{\frac{m}{n}}$ is real if $a<0$.
- It depends on the root index (denominator) and on how the fraction is represented.
- Restricting to $a>0$ avoids contradictions.
Decimal Exponents Can Often Be Rewritten as Rational Exponents
Many decimal exponents are actually fractions in disguise.
- Finite decimals are rational numbers. Example: $0.75=\frac{75}{100}=\frac{3}{4}$.
- Some repeating decimals are rational numbers. Example: $0.\dot{3}=\frac{1}{3}$.
So you can rewrite a decimal exponent as a rational exponent and then as a radical.
$$5^{0.75}=5^{\frac{3}{4}}=\sqrt[4]{5^3}$$
$$5^{0.1}=5^{\frac{1}{10}}=\sqrt[10]{5}$$
$$5^{1.6}=5^{\frac{8}{5}}=\sqrt[5]{5^8}$$
- To convert a terminating decimal to a fraction, write it over a power of 10 and simplify.
- For a repeating decimal, use an algebra trick (for example, let $x=0.\dot{3}$, then $10x=3.\dot{3}$, subtract to get $9x=3$).
When simplifying expressions with radicals and rational exponents, a consistent strategy reduces errors.
- Rewrite radicals as rational exponents (or rewrite exponents as radicals) so everything uses one notation.
- Rewrite numbers as prime powers when possible, so bases match.
- Use the index laws to add, subtract, or multiply exponents.
- Convert back to a simplified radical form if the question asks for it.
- Rewrite $\sqrt[3]{x^2}$ using exponents.
- Simplify $x^{\frac{1}{2}}\cdot x^{\frac{2}{3}}$ as a single power of $x$.
- Evaluate $8^{\frac{2}{3}}$ without a calculator.
