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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.

Cubing a number gives the volume of a cube


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}$3125 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}$364.

ThinkWe should read $\sqrt[3]{64}$364 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$364=4.


Practice questions


Evaluate $\sqrt[3]{27}$327


Solve $x^3=64$x3=64.



Use square root symbols to represent solutions to equations of the form x^2 = p, where p is a positive rational number.

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