Worked examples
The question
Question: work out 25^(1/2).
25^(1/2)
A fraction power has two jobs.
The bottom number, 2, says which root: the square root.
The top number, 1, says which power.
For the square root, the small 2 is not written.
a^(m/n) = (n√a)m
Careful!
The power is not times.
25 times 1/2 is a different number.
Careful: not 25^(1/2) = 25 × 12 ✗
1 Write it as a root
The bottom is 2, so it is the square root of 25.
The top is 1: power 1 changes nothing.
= √25
√25 is 5, because 5 × 5 is 25.
5 × 5 = 25
So 25^(1/2) = 5.
= 5
2 Check
Let's check: 5 times itself 2 times.
We get 25.
So the answer is right!
5 × 5 = 25
Let's look back.
First: write it as a root.
Last: check.
▶ Watch it solved
The question
Question: work out 8^(2/3).
8^(2/3)
A fraction power has two jobs.
The bottom number, 3, says which root: the cube root.
The top number, 2, says which power.
a^(m/n) = (n√a)m
Careful!
The power is not times.
8 times 2/3 is a different number.
Careful: not 8^(2/3) = 8 × 23 ✗
1 Take the root
The bottom is 3: the cube root of 8.
The top is 2: to the power 2.
We take the root first.
It keeps the numbers small.
= (3√8)2
∛8 is 2, because 2 × 2 × 2 is 8.
2 × 2 × 2 = 8
We write 2 in place of ∛8.
= 22
2 Take the power
2 to the power 2 is 2 × 2.
That is 4.
2 × 2 = 4
So 8^(2/3) = 4.
= 4
3 Check
Let's check the other way: the power first, then the root.
8 to the power 2 is 64.
Other order: 82 = 64
The cube root of 64 is 4, because 4 × 4 × 4 is 64.
We get 4 again.
But the numbers were bigger.
So root first is easier.
And the answer is right!
3√64 = 4, 4 × 4 × 4 = 64
Let's look back.
First: take the root.
Then: take the power.
Last: check.
▶ Watch it solved
The question
Question: work out 9^(−1/2).
9^(−1/2)
A fraction power has two jobs.
The bottom number, 2, says which root: the square root.
The top number, 1, says which power.
For the square root, the small 2 is not written.
a^(m/n) = (n√a)m
Careful!
A minus in the power does not make the answer negative.
It means 1 over.
Careful: not 9^(−1/2) = −3 ✗
1 Remove the minus
A minus power means 1 over the same power without the minus.
a−n = 1an
So 9^(−1/2) is 1/9^(1/2).
= 19^(1/2)
2 Write it as a root
The bottom is 2, so it is the square root of 9.
The top is 1: power 1 changes nothing.
= 1√9
√9 is 3, because 3 × 3 is 9.
3 × 3 = 9
So 9^(−1/2) = 1/3.
= 13
3 Check
Let's check.
Without the minus, 9^(1/2) is 3.
A power times its minus power is 9 to the power 0, which is 1.
9^(1/2) = 3, 90 = 1
3 times 1/3 is 1.
So the answer is right!
3 × 13 = 1
Let's look back.
First: remove the minus.
Then: write it as a root.
Last: check.
▶ Watch it solved
