Finding the cube root of a number is the reverse of cubing a number. We express the cube root using the radical symbol
For example, the "cube root of " equals since
So, we write
Some basic cube roots are as follows:
A perfect cube is a number whose cube root is a whole number. So, the numbers
are all perfect cubes. You need to remember them.
What is
Since the number under the cube root () is already written as a cube, we obtain the answer instantly:
$\sqrt[3]{(-75)^3}=$
|
a
|
$-75$ |
|
b
|
$-5$ |
|
c
|
$25$ |
|
d
|
$-15$ |
|
e
|
$75$ |
$\sqrt[3]{\left(\dfrac{1}{5}\right)^3}=$
|
a
|
$\dfrac{1}{125}$ |
|
b
|
$5$ |
|
c
|
$\dfrac{1}{5}$ |
|
d
|
$\dfrac{1}{15}$ |
|
e
|
$\dfrac{1}{25}$ |
Calculate the value of
First, notice that the numbers and are both perfect cubes.
So, to determine the value of we express as a perfect cube:
$\sqrt[3]{125} = $
|
a
|
$15$ |
|
b
|
$-5$ |
|
c
|
$25$ |
|
d
|
$5$ |
|
e
|
$-25$ |
$\sqrt[3]{\dfrac{1}{8}}=$
|
a
|
$\dfrac{1}{24}$ |
|
b
|
$\dfrac{3}{2}$ |
|
c
|
$\dfrac{1}{2}$ |
|
d
|
$\dfrac{1}{4}$ |
|
e
|
$\dfrac{3}{8}$ |
$\sqrt[3]{\dfrac{27}{64}}=$
|
a
|
$\dfrac{9}{4}$ |
|
b
|
$\dfrac{3}{2}$ |
|
c
|
$\dfrac{3}{16}$ |
|
d
|
$\dfrac{3}{4}$ |
|
e
|
$\dfrac{9}{8}$ |
Unlike the square root, it is possible to find the cube root of a negative number.
For example, because
Some basic cube roots of negative numbers are as follows:
Note that the numbers
are all perfect cubes.
Calculate
First, notice that the number is a perfect cube.
So, to determine the value of we express as a perfect cube:
$\sqrt[3]{-8}=$
|
a
|
$-8$ |
|
b
|
$-4$ |
|
c
|
$-2$ |
|
d
|
$4$ |
|
e
|
$2$ |
$\sqrt[3]{-\dfrac{1}{8}} = $
|
a
|
$\dfrac{2}{3}$ |
|
b
|
$\dfrac{1}{2}$ |
|
c
|
$-\dfrac{3}{2}$ |
|
d
|
$-\dfrac{1}{4}$ |
|
e
|
$-\dfrac{1}{2}$ |
$\sqrt[3]{-\dfrac{125}{8}}=$
|
a
|
$\dfrac{5}{4}$ |
|
b
|
$-\dfrac{4}{5}$ |
|
c
|
$-\dfrac{5}{2}$ |
|
d
|
$\dfrac{25}{2}$ |
|
e
|
$-\dfrac{5}{4}$ |
Let's calculate by performing each operation one step at a time: Notice that the cubing the cube root gives the number under the cube root:
This result holds in general. If is any number, then
The cube and the cube root cancel each other out, and we're left with the number that we started with.
Find the value of
The cube and the cube root cancel each other out. So, we have
$(\sqrt[3]{125})^3=$
|
a
|
$5$ |
|
b
|
$-12$ |
|
c
|
$-125$ |
|
d
|
$-5$ |
|
e
|
$125$ |
$(\sqrt[3]{-125})^3=$
|
a
|
$5$ |
|
b
|
$-5$ |
|
c
|
$3$ |
|
d
|
$125$ |
|
e
|
$-125$ |
We can think of the cube root geometrically. For instance, let's represent the number by a collection of eight items:
Now, we can pile them up in the form of a cube where each side is equal to
