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1.05 Cube roots of perfect cubes

Lesson

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.

Cubing a number gives 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, $$3125 represents the cube root of $$125 which is equivalent to $$5 because $$5×5×5=125.

 

Worked example

Question 1

Evaluate $$364.

ThinkWe should read $$364 as "the cube root of $$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 $$364=4.

 

Practice questions

QUESTION 2

Evaluate $$327

QUESTION 3

Consider $$x3=64.

  1. Complete the rearranged equation:

    $$x$$=$$3

  2. Solve the equation for $$x.

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