Cube Root
Cube root is a Grade 7 math skill from Yoshiwara Intermediate Algebra introducing the cube root function and notation. The cube root of x (written as the cube root of x) gives the number whose cube equals x, and unlike square roots, cube roots are defined for all real numbers including negatives.
Key Concepts
Property $b$ is the cube root of $a$ if $b$ cubed equals $a$. In symbols, we write $$b = \sqrt[3]{a} \quad \text{if} \quad b^3 = a$$ Unlike square roots, which are not real for negative numbers, every real number has a real cube root. Simplifying radicals occurs at the same level as powers in the order of operations.
Examples To simplify $3\sqrt[3]{ 8}$, we find the cube root of $ 8$ which is $ 2$, and then multiply by $3$. So, $3\sqrt[3]{ 8} = 3( 2) = 6$. To evaluate $2 \sqrt[3]{ 125}$, we first find that the cube root of $ 125$ is $ 5$. The expression becomes $2 ( 5) = 7$. To simplify $\frac{10 + \sqrt[3]{ 27}}{7}$, we calculate $\sqrt[3]{ 27} = 3$. The expression becomes $\frac{10 + ( 3)}{7} = \frac{7}{7} = 1$.
Explanation A cube root is the inverse operation of cubing a number. Think of it as asking: 'What number, when multiplied by itself three times, gives me this value?' Unlike square roots, you can take the cube root of negative numbers.
Common Questions
What is a cube root?
The cube root of x is the number y such that y^3 = x. For example, the cube root of 27 is 3, since 3^3 = 27.
Is the cube root of a negative number real?
Yes. The cube root of -8 is -2, since (-2)^3 = -8. Odd roots of negative numbers are always real.
How does cube root differ from square root?
The square root of a negative number is not real, but the cube root of a negative number is always real because odd powers preserve the sign.
What is the cube root of 0?
The cube root of 0 is 0, since 0^3 = 0.