Square and Cube Roots

In this lesson we will study some ways to think about second and third powers by using geometric figures. We will also learn about square roots and cube roots, which reverse the operations of taking the second and third power of a number.

Squares and square roots

Raising a number to its second power is called squaring that number. For example, the square of 3 is $3^2 = 3(3) = 9$. This is because a square whose sides are each 3 units long has an area (number of small green squares) of 9 square units, as shown on the grid to the left.

Find the square of each number in the table below. Click the Next button in each row to draw that row’s problem on the grid to the left.

Number
($x$)		The square of
that number ($x^2$)

Squaring a positive number (like 3) answers a question like: “What is the area of a square whose sides are 3 units long?” Sometimes we might have questions about squares that go in the opposite direction, like: “How long are the sides of a square whose area is 9 square units?” As you have seen, its sides will each be 3 units long, because 3 is a positive number and $3^2=9$.

How long are the sides of a square whose area is 4 square units? Click to see a helpful picture; you may also use your answers from square-qn. units
Click . How long are the sides of a square whose area is 16 square units? units

Because 3 is a non-negative number whose square is 9, 3 is called the square root of 9. The symbol $√\phantom{3}$ is used for square roots in mathematical expressions and equations. That is, the way to write “3 is the square root of 9” as an equation is: $3=√9$.

Find the square root of each number in the table below. Each problem is pictured on the grid to the left.

Number
($x$)		The square root of
that number ($√x$)

You have seen that $√9=3$ and $√16=4$. Let’s try to figure out the value of $√10$.

A square whose area is 10 square units is drawn on the grid to the left. This means that the length of each of its sides is $√10$. Is this length more than 3, less than 3, or equal to 3?
Is it more than 4, less than 4, or equal to 4?
Are there any whole numbers between 3 and 4?
Is $√10$ a whole number?

Using a calculator, find the square of each number in the table below, and note whether it is more or less than 10.

Number
($x$)		Square of
number ($x^2$) 		Is $x^2$ more
or less than 10?

From this table, you can see that $√10$ is some number between 3.16 and 3.17, but we still don’t have an exact decimal value for it. Whenever a square root of a whole number is not a whole number, this is what happens. We can approximate the square root as well as we want if we’re willing to work hard enough, but we can’t give an exact decimal value.

Cubes and cube roots

The expression $x^2$ is called the square of $x$ because it is the area of a square whose sides are $x$ units long. Now we’ll look at the expression $x^3$, to see if it has a similar geometric meaning.

The figure to the left consists of 5 square slices, each of which contains $5^2$ unit cubes (cubes with edge length 1). How many unit cubes are in the figure in total? (You can rotate the figure by dragging it to the left or right.)
You can slide the slider to the left to move the slices closer together. What shape (pyramid, cube, sphere...) is formed when the slices touch each other (that is, when $d=0$)?

The figure to the left demonstrates that a cube whose edges are 5 units long has a volume of $5^3$ cubic units, because it contains that many unit cubes. Similarly, a cube whose edges are any positive number $x$ long has a volume of $x^3$. For this reason, raising a number to the third power is called cubing that number or finding its cube.

Find the cube of each number in the table below. Click the Next button in each row to show that row’s problem to the left.

Number
($x$)		The cube of
that number ($x^3$)

Cubing a positive number (like 2) answers a question like: “What is the volume of a cube whose edges are 2 units long?” Sometimes we might have questions about cubes that go in the opposite direction, like: “How long are the edges of a cube whose volume is 8 cubic units?” As you have seen, its edges will each be 2 units long, because 2 is a positive number and $2^3=8$.

How long are the edges of a cube whose volume is 27 cubic units? Click to see a helpful picture; you may also use your answers from cube-table-qn. units

Because 2 is a number whose cube is 8, 2 is called the cube root of 8. The symbol $√^3\phantom{2}$ is used for cube roots in mathematical expressions and equations. That is, the way to write “2 is the cube root of 8” as an equation is: $2=√^3{8}$.

Find the cube root of each number in the table below. Each problem is pictured on the grid to the left.

Number
($x$)		The cube root of
that number ($√^3{x}$)
The cube drawn to the left has a volume of 15 cubic units, which means that its edges are $√^3{15}$ units long. Is $√^3{15}$ more than 2, less than 2, or equal to 2?
Is $√^3{15}$ more than 3, less than 3, or equal to 3?
Remember that there are no whole numbers between 2 and 3. Is $√^3{15}$ a whole number?

Using a calculator, find the cube of each number in the table below, and note whether it is more or less than 15.

Number
($x$)		Cube of
number ($x^3$) 		Is $x^3$ more
or less than 15?

The situation with whole numbers whose cube roots are not whole numbers is a lot like the situation with square roots that are not whole numbers. We can approximate the cube roots as well as we want, but we cannot find an exact decimal value.

Roots and negative numbers

So far, we’ve mostly studied square and cube roots of positive numbers. Now let’s look at square and cube roots of negative numbers.

What is $(-2)^2$? It is drawn on the grid to the left.
Is $(-2)^2$ positive or negative?

The square of any nonzero number, either positive or negative, is positive. So there’s no real number whose square is $-4$.

Does $-4$ have a square root?
What is $(-2)^3$? Click to see it drawn to the left.
Is $(-2)^3$ positive or negative?
Does $-8$ have a cube root?
What is the cube root of $-8$?

In general:

Negative numbers do not have real square roots. The cube root of a negative number $-a$ is $-√^3{a}$.

Find the cube root of each negative number in the table below. Each problem is pictured on the grid to the left.

Number
($x$)		The cube root of
that number ($√^3{x}$)