 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 $x^3$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

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{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{64}$364.

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

Practice questions

QUESTION 2

Evaluate $\sqrt{27}$327

QUESTION 3

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

Outcomes

8.NS.4

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