Surds

UK Secondary (7-11)

Evaluate expressions involving numerical roots

Lesson

Finding the square root of a number is the opposite operation to squaring a number. Similarly, finding the cube root of a number is the opposite operation to cubing a number. We discussed squaring a cubing numbers in Shaping Numbers which you can look over if you need a reminder.

If we are asked to find the square root of a value, we are being asked, "What number multiplied by itself would give this value?"

You might also see the square root symbol written with a number inside it. For example, $\sqrt{25}$√25. This means. "Find the square root of $25$25."

Careful!

If you try to find the square root of a negative number in your calculator, you'll get an error message. This is because the multiplying two real negative numbers produces a positive answer.

e.g. We cannot find $\sqrt{-9}$√−9. Click here if you need a refresher on multiplying negative numbers.

Evaluate: $\sqrt{64}$√64

Think: $8\times8=64$8×8=64

Do: $\sqrt{64}=8$√64=8

If we are asked to find the cube root of a value, we are being asked, "What number multiplied by itself twice would give this value?"

It's most common to see questions written with the cube root symbol, such as $\sqrt[3]{225}$^{3}√225.

Think

You *can* find the cubed root of a negative number. Think of why it is different to squaring a number. Click here if you need a refresher on multiplying negative numbers.

Evaluate:** **$\sqrt[3]{64}$^{3}√64

Think: $4\times4\times4=64$4×4×4=64

Do:** **$\sqrt[3]{64}=4$^{3}√64=4

Evaluate: $\sqrt[3]{-27}$^{3}√−27.

Think: What number multiplied by itself *twice* would equal $-27$−27?

Do: Since we have a negative number under the cube root sign, we know our answer will be negative.

$-3\times\left(-3\right)\times\left(-3\right)=-27$−3×(−3)×(−3)=−27

So $\sqrt[3]{-27}=-3$^{3}√−27=−3

Now let's look at putting all this knowledge together in different types of questions. Problems involving square or cube roots may involve fractions, so it's good to be familiar with multiplying and dividing fractions.

Evaluate $\sqrt{\frac{16}{100}}$√16100.

Leave your answer as a fraction.

Evaluate $\frac{-1}{\sqrt{\frac{16}{100}}}$−1√16100