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 $x^3$x3 has the following meanings:

$x^3$x3

a number $x$x raised to the power of three

$x$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$8 is a perfect cube because it can be expressed as $2\times2\times2$2×2×2 or $2^3$23.

$1$1	$=$=	$1\times1\times1$1×1×1	$=$=	$1^3$13
$8$8	$=$=	$2\times2\times2$2×2×2	$=$=	$2^3$23
$27$27	$=$=	$3\times3\times3$3×3×3	$=$=	$3^3$33
$64$64	$=$=	$4\times4\times4$4×4×4	$=$=	$4^3$43
$125$125	$=$=	$5\times5\times5$5×5×5	$=$=	$5^3$53
$216$216	$=$=	$6\times6\times6$6×6×6	$=$=	$6^3$63
$343$343	$=$=	$7\times7\times7$7×7×7	$=$=	$7^3$73
$512$512	$=$=	$8\times8\times8$8×8×8	$=$=	$8^3$83
$729$729	$=$=	$9\times9\times9$9×9×9	$=$=	$9^3$93
$1000$1000	$=$=	$10\times10\times10$10×10×10	$=$=	$10^3$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, $\sqrt[3]{125}$³√125 represents the cube root of $125$125 which is equivalent to $5$5 because $5\times5\times5=125$5×5×5=125.

 

Worked example

Question 1

Evaluate $\sqrt[3]{64}$³√64.

Think: We should read $\sqrt[3]{64}$³√64 as "the cube root of $64$64".

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

We know that $64=4\times4\times4$64=4×4×4.

Do: This means the cube root of $64$64 is $4$4, so $\sqrt[3]{64}=4$³√64=4.

 

Practice questions

QUESTION 2

QUESTION 3