maths > exponents

Representation of Cubes and Cube Roots

what you'll learn...

overview

In this page, cube of a number and cube root of a number are revised.

cube

${5}^{3}=5×5×5=125$${5}^{3} = 5 \times 5 \times 5 = 125$?

An exponent of power $3$$3$ is called cube of the number.

eg: Cube of $7$$7$ is ${7}^{3}=343$${7}^{3} = 343$.

The word "cube" means : the 3D shape of equal sides and ${90}^{\circ }$${90}^{\circ}$ angles" between edges. The exponent cube is representative of "volume of the shape cube, side to the power 3".

cube root

${8}^{\frac{1}{3}}={\left(2×2×2\right)}^{\frac{1}{3}}=2$${8}^{\frac{1}{3}} = {\left(2 \times 2 \times 2\right)}^{\frac{1}{3}} = 2$
$\sqrt[3]{8}=\sqrt[3]{2×2×2}=2$$\sqrt[3]{8} = \sqrt[3]{2 \times 2 \times 2} = 2$

An exponent of power $\frac{1}{3}$$\frac{1}{3}$ or the $3$$3$rd root is called cube root of the number.

eg: Cube root of $64$$64$ is ${64}^{\frac{1}{3}}=\sqrt[3]{64}=4$${64}^{\frac{1}{3}} = \sqrt[3]{64} = 4$.

examples

Cube of $3$$3$ is ${3}^{3}$${3}^{3}$.

Cube root of $8$$8$ is $\sqrt[3]{8}={8}^{\frac{1}{3}}$$\sqrt[3]{8} = {8}^{\frac{1}{3}}$.

finding cube root

How to find $\sqrt[3]{216}$$\sqrt[3]{216}$?

Cube root is a form of roots. In roots, we learned to perform prime factorization to find the root.
$\sqrt[3]{216}$$\sqrt[3]{216}$
$=\sqrt[3]{2×2×2×3×3×3}$$= \sqrt[3]{2 \times 2 \times 2 \times 3 \times 3 \times 3}$
$=2×3$$= 2 \times 3$
$=6$$= 6$

summary

Cube of a number : A number to the power $3$$3$ is the cube of the number.
eg: ${6}^{3}=6×6×6=216$${6}^{3} = 6 \times 6 \times 6 = 216$

Cube Root of a number : A number to the power $\frac{1}{3}$$\frac{1}{3}$ is the cube root of the number.
eg: $\sqrt[3]{216}=6$$\sqrt[3]{216} = 6$ as ${6}^{3}=216$${6}^{3} = 216$.

Finding Cube Roots : To find cube root of a number, express the number in prime factors and group the factors.

eg: $\sqrt[3]{1000}$$\sqrt[3]{1000}$ $=\sqrt[3]{2×2×2×5×5×5}$$= \sqrt[3]{2 \times 2 \times 2 \times 5 \times 5 \times 5}$ $=2×5=10$$= 2 \times 5 = 10$

Note: This method is suitable for finding cube roots resulting in integers.

Outline

The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

→   Representation of Exponents

→   Inverse of exponent : root

→   Inverse of exponent : Logarithm

→   Common and Natural Logarithms

→   Exponents Arithmetics

→   Logarithm Arithmetics

→   Formulas

→   Numerical Expressions

→   PEMA / BOMA

→   Squares and Square roots

→   Cubes and Cube roots