1.03 Square roots and cube roots
Square Numbers
We can represent repeated addition using multiplication, for example:
$3+3+3+3=4\times3$3+3+3+3=4×3
In a similar way, we can represent repeated multiplication using powers.
To indicate that a number has been multiplied by itself we write the number being multiplied, called the base, and then indicate the number of times it is multiplied by itself by writing this number, called the power (also known as the index or exponent), above and to the right of the base. Here is an example:
$3\times3=3^2$3×3=32
This is read as "three to the power of two" or "three raised to the power of $2$2".
For now we will just be looking at numbers raised to the power of two, known as square numbers or perfect squares. In the above example $3^2$32 is often referred to as "three squared".
What does it mean to 'square' a number? Look at the pattern below!
| $1^2$12 | $2^2$22 | $3^2$32 | $4^2$42 | $5^2$52 |
|---|---|---|---|---|
 |
 |
 |
 |
 |
| 1 | 4 | 9 | 16 | 25 |
You can see above that each formation of dots forms a square.
| $1^2$12 | $=$= | $1\times1$1×1 | $=$= | $1$1 | |
| $2^2$22 | $=$= | $2\times2$2×2 | $=$= | $4$4 | |
| $3^2$32 | $=$= | $3\times3$3×3 | $=$= | $9$9 | |
| $4^2$42 | $=$= | $4\times4$4×4 | $=$= | $16$16 | and so on. |
Use the applet below to explore the first twelve square numbers.
Finding the square root
If we are asked to find the square root of a value, we are being asked, "What number multiplied by itself would give this value?"
You might also see the square root symbol written with a number inside it, for example, $\sqrt{25}$√25.
This means find the square root of $25$25.
Worked example 1
What does the value of $\sqrt{144}$√144 equal?
Think: We should read $\sqrt{144}$√144 as "the square root of $144$144".
This is the number that squares (multiplies with itself) to make $144$144.
We know that $144=12\times12$144=12×12.
Do: The square root of $144$144 is $12$12.
Practice questions
QUESTION 1
Evaluate $9^2$92
QUESTION 2
Evaluate $4^2+9^2$42+92
QUESTION 3
Evaluate $\sqrt{8^2+6^2}$√82+62
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[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.
In a similar way we can also calculate the fourth root and fifth root.
Worked example
example 2
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.
EXAMPLE 3
Evaluate $\sqrt[4]{16}$4√16.
Think: We should read $\sqrt[4]{16}$4√16 as "the fourth root of $16$16".
This is the number multiplied by itself four times to make $16$16.
We know that $16=2\times2\times2\times2$16=2×2×2×2.
Do: This means the fourth root of $16$16 is $2$2, so $\sqrt[4]{16}=2$4√16=2.
EXAMPLE 4
Evaluate $\sqrt[5]{243}$5√243.
Think: We should read $\sqrt[5]{243}$5√243 as "the fifth root of $243$243".
This is the number multiplied by itself five times to make $243$243.
You can use trial and error to find out that $243=3\times3\times3\times3\times3$243=3×3×3×3×3.
Do: This means the fifth root of $243$243 is $3$3, so $\sqrt[5]{243}=3$5√243=3.
Practice questions
QUESTION 4
Evaluate $\sqrt[3]{27}$3√27
QUESTION 5
Consider $x^3=64$x3=64.
Complete the rearranged equation:
$x$x$=$=$\sqrt[3]{\editable{}}$3√
Solve the equation for $x$x.