EXT: Evaluate expressions involving numerical roots

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.

Finding the square root

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."

Example

Question 1

Evaluate: $\sqrt{64}$√64

Think: $8\times8=64$8×8=64

Do: $\sqrt{64}=8$√64=8 

 

Finding the cube root

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.

Examples

question 2

Evaluate: $\sqrt[3]{64}$³√64

Think: $4\times4\times4=64$4×4×4=64

Do: $\sqrt[3]{64}=4$³√64=4 

 

question 3

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

 

Solving problems with square roots and cube roots

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.

Examples

Question 4

Question 5