2.1 Laws of indices for rational exponents — Learn

Prerequisites: integer powers; square roots and cube roots; multiplying and dividing fractions; the index laws for whole-number exponents.

By the end of this section you can: write a root as a power; evaluate an expression such as $16^{\frac{3}{4}}$; apply the index laws with fractional and negative exponents; and solve an equation in which the unknown is a base with a rational exponent.

1. The index laws you already know

An index (or power) counts repeated multiplication. For a base $a$ and whole numbers $m$ and $n$:

These laws work for every exponent, not only whole numbers. Two more follow from them.

The second law turns a negative index into a reciprocal. For example, $5^{-2} = \frac{1}{25}$ and $\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}$.

2. A root is a fractional power

The square root of $a$ is the number whose square is $a$. We write $\sqrt{a} = a^{\frac{1}{2}}$, because

In the same way, the $n$th root is the power $\frac{1}{n}$:

A power with a numerator builds on this. The numerator counts how many times the root is repeated, and the denominator names the root:

For example, $8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$. When you evaluate by hand, take the root first and raise the result to the power. The numbers stay small that way.

Example 1. Work out $16^{\frac{3}{4}}$.

Take the fourth root first:

The fourth root of $16$ is $2$, because $2^4 = 16$. Raise $2$ to the power $3$:

Example 2. Work out $125^{-\frac{1}{3}}$.

The negative sign gives a reciprocal:

The cube root of $125$ is $5$:

3. The laws in algebra

The same laws simplify expressions with a letter base. Add the exponents when you multiply, subtract when you divide, and multiply when a power is raised to a power.

Example 3. Simplify $x^{\frac{1}{2}} \times x^{\frac{3}{2}}$, where $x > 0$.

Add the exponents:

Example 4. Simplify $\left(8x^6\right)^{\frac{2}{3}}$, where $x > 0$.

The power applies to each factor inside the bracket:

Now $8^{\frac{2}{3}} = 4$, and the power of a power multiplies the exponents:

4. Solving an equation with a rational exponent

To free an unknown from an exponent, raise both sides to the reciprocal power.

Example 5. Solve $x^{\frac{2}{3}} = 9$.

The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. Raise both sides to that power:

The left side becomes $x^1 = x$, and the right side becomes $27$:

There is a second solution. The exponent $\frac{2}{3}$ has an even numerator, so a negative base gives the same value:

So $x = 27$ or $x = -27$. Check both values in the original equation before you finish.

Watch out. Three traps appear with rational exponents.

Splitting a root over a sum. The law $a^{\frac{1}{n}} b^{\frac{1}{n}} = (ab)^{\frac{1}{n}}$ joins a product, not a sum. $\sqrt{9 + 16} = 5$, but $\sqrt{9} + \sqrt{16} = 7$, so $\sqrt{a + b}$ is not $\sqrt{a} + \sqrt{b}$.

A negative index and a negative number. $2^{-3}$ means $\frac{1}{2^3} = \frac{1}{8}$; it does not mean $-8$. The minus sign in the exponent moves the base to the other part of a fraction.

A lost root. Raising both sides to an even power can hide a second solution, and raising to the power $\frac{3}{2}$ needs a non-negative base. Re-check: after solving, substitute each value back, and ask whether the base is allowed to be negative.

Progress check

1. Write $\sqrt[3]{x}$ as a power of $x$.
2. Work out $81^{\frac{3}{4}}$ by taking the root first.
3. Solve $x^{\frac{3}{2}} = 8$.
4. Explain why $x^{\frac{1}{2}}$ has no real value when $x = -4$.

You should now be able to

• write a root as a fractional power, and a fractional power as a root;
• evaluate a power with a fractional or negative exponent;
• apply the index laws to expressions with letter bases;
• solve an equation of the form $x^{\frac{m}{n}} = k$ and find every real solution;
• explain why a root cannot be split across a sum.