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Indices: Fractional Powers

# Indices: Fractional Powers

### Indices: Fractional Powers

Fractional indices are like finding roots, while negative indices are like finding reciprocals. We can also combine these in more difficult scenarios.

1

The power in an index number is not always an integer.

$x^{\frac{1}{2}}$ is an example of an index number with a fractional power.

2

To understand how fractional powers work, let's consider $x^{\frac{1}{2}}$ and multiply it by itself. What is $x^{\frac{1}{2}}\times x^{\frac{1}{2}}$?

3

When we multiply a number by itself, we get a square, for example $a\times a=a^2$. In this example what is $a$?

4

In our example, we had $x^{\frac{1}{2}}\times x^{\frac{1}{2}}=x$.

This means that $x^{\frac{1}{2}}$ is the square root of $x$ or $x^{\frac{1}{2}}=\sqrt{x}$.

1

Let's have a look at the index number with fractional power $\frac{1}{3}$

Our example is $y^{\frac{1}{3}}$.

2

This time, work out $y^{\frac{1}{3}} \times y^{\frac{1}{3}}\times y^{\frac{1}{3}}$.

3

What is a number that is multiplied by itself three times, for example $a\times a\times a$?

4

We know that $y^{\frac{1}{3}} \times y^{\frac{1}{3}}\times y^{\frac{1}{3}}=y$

This means that $y^{\frac{1}{3}}$ is the cube root of $y$ or $y^{\frac{1}{3}}=\sqrt[3]{y}$.

1

We can see a pattern here $x^{\frac{1}{2}}=\sqrt{x}$

$y^{\frac{1}{3}}=\sqrt[3]{y}$.

The general rule, a fractional power with numerator $1$, is the root of the base number of the same value as the denominator.

2

What is the meaning of $a^{\frac{1}{6}}$?

3

We can work out some values. What is $16^{\frac{1}{2}}$?

Let's try another one, what is $27^{\frac{1}{3}}$?

We can apply this to other fractional powers:

$64^\frac{1}{3}=\sqrt[3]{64}=4$

$625^{\frac{1}{4}}=\sqrt[4]{625}=5$

What is $32^{\frac{1}{5}}$?

We can also use larger fractional powers, where the numerator is more than one.

Let's try an example: $8^{\frac{2}{3}}$

1

We can split the fractional power $\frac{2}{3}$ into two parts as $\frac{2}{3} = \frac{1}{3} \times 2$. How can we show this using brackets?

2

We can now work out the answer is two parts. First, what is the result of $8^{\frac{1}{3}}$?

3

Replace $8^{\frac{1}{3}}$ with $2$ in $(8^{\frac{1}{3}})^2$.

$(2)^2$

4

Find$(2)^2$.

5

Nice! 👍

The answer is $8^{\frac{2}{3}}=4$.

Let's try a slightly harder one. What is $64^{-\frac{2}{3}}$?

1

We can generalise the rule

Notice that there is a negative in the power too, but we can use the rule in the same way.

2

The power in $64^{-\frac{2}{3}}$ is $\frac{-2}{3}$ , how can we split this down to use in the rule?

3

We now have $64^{-\frac{2}{3}}=(64^{\frac{1}{3}})^{-2}$. Let's work out what is in the brackets, what is $64^{\frac{1}{3}}$?

4

We can now replace $64^{\frac{1}{3}}$ in $(64^{\frac{1}{3}})^{-2}$ with $4$.

This gives $(4)^{-2}$.

5

Finally we work out $(4)^{-2}$. Take care with the negative sign - this is asking you to take the reciprocal of the positive power.

6

You have shown that $64^{-\frac{2}{3}}=\dfrac{1}{16}$

Well done! 👍

Final question: what is the value of $27^\frac{2}{3}$?

Summary!

1

A fractional power with a numerator of $1$ is the same as the root of the same value as the denominator

$x^{\frac{1}{2}}=\sqrt{x}$

$y^{\frac{1}{3}}=\sqrt[3]{y}$

2

A fractional power with a numerator greater than $1$

is the root of the base number of the same value as the denominator then raised to the power of the same value as the numerator.