Exponents : Representation of Cubes and Cube Roots

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

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

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 $90^{\circ}$ $90^@$ angles" between edges. The exponent cube is representative of "volume of the shape cube, side to the power 3".

cube root

$8^{\frac{1}{3}} = {(2 \times 2 \times 2)}^{\frac{1}{3}} = 2$ $8^(1/3) = (2 xx 2 xx 2)^(1/3) = 2$
$\sqrt[3]{8} = \sqrt[3]{2 \times 2 \times 2} = 2$ $root(3)(8) = root(3)(2 xx 2 xx 2) = 2$

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

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

examples

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

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

finding cube root

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

Cube root is a form of roots. In roots, we learned to perform prime factorization to find the root.
$\sqrt[3]{216}$ $root(3)(216)$
$= \sqrt[3]{2 \times 2 \times 2 \times 3 \times 3 \times 3}$ $=root(3)(2xx2xx2xx3xx3xx3)$
$= 2 \times 3$ $=2xx3$
$= 6$ $=6$

summary

Cube of a number : A number to the power $3$ $3$ is the cube of the number.
eg: $6^{3} = 6 \times 6 \times 6 = 216$ $6^3 = 6xx6 xx 6 = 216$

Cube Root of a number : A number to the power $\frac{1}{3}$ $1/3$ is the cube root of the number.
eg: $\sqrt[3]{216} = 6$ $root(3)(216) = 6$ 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: $\sqrt[3]{1000}$ $root(3)(1000)$ $= \sqrt[3]{2 \times 2 \times 2 \times 5 \times 5 \times 5}$ $=root(3)(2xx2xx2xx5xx5xx5)$ $= 2 \times 5 = 10$ $=2xx5 =10$

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