1.05 Cube roots of perfect cubes

Recall that raising a number to the power of three is often called "cubing" a number. For example, the expression $$x3 has the following meanings:

$$x3

a number $$x raised to the power of three

$$x cubed

Just as the square of a number relates to the area of a square, cubing a number relates to the volume of a cube.

Exploration

Let's look at a table of the first ten perfect cubes. A perfect cube is a number that can be expressed as the cube of an integer. For example, the number $$8 is a perfect cube because it can be expressed as $$2×2×2 or $$23.

$$1	$$=	$$1×1×1	$$=	$$13
$$8	$$=	$$2×2×2	$$=	$$23
$$27	$$=	$$3×3×3	$$=	$$33
$$64	$$=	$$4×4×4	$$=	$$43
$$125	$$=	$$5×5×5	$$=	$$53
$$216	$$=	$$6×6×6	$$=	$$63
$$343	$$=	$$7×7×7	$$=	$$73
$$512	$$=	$$8×8×8	$$=	$$83
$$729	$$=	$$9×9×9	$$=	$$93
$$1000	$$=	$$10×10×10	$$=	$$103

 

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 three times would give this value?"

We might also see the cube root symbol written with a number inside it, for example, $$³√125 represents the cube root of $$125 which is equivalent to $$5 because $$5×5×5=125.

 

Worked example

Question 1

Evaluate $$³√64.

Think: We should read $$³√64 as "the cube root of $$64".

This is the number multiplied by itself three times to make $$64.

We know that $$64=4×4×4.

Do: This means the cube root of $$64 is $$4, so $$³√64=4.

 

Practice questions

QUESTION 2

QUESTION 3