Question

What is the cube root of $ 216{x^9}{y^{18}} $ ?

Answer 1

Hint: Here, first of all will re-write the terms in the form of factors in the cube and then will apply the properties of the power and exponent and then simplify the expression for the required value.

Complete step-by-step answer:
words, cube-root of a number is the factor which is multiplied by it for three times to get that number. It is expressed as $ \sqrt[3]{n} $ and is read as the cube root of the number “n”. For example $ \sqrt[3]{{27}} = \sqrt[3]{{3 \times 3 \times 3}} = 3 $ since here, the number “ $ 3 $ “ is repeated three times.
Given expression: $ \sqrt[3]{{216{x^9}{y^{18}}}} $
Find the factors and find the cube root of the constant term. Replace $ {6^3} = 216 $
 $ = \sqrt[3]{{{6^3}{x^9}{y^{18}}}} $
The above expression can be re-written as –
 $ = {({6^3}{x^9}{y^{18}})^{\dfrac{1}{3}}} $
By the property of the Power rule: to raise Power to power you have to multiply the exponents such as - $ {\left( {{x^a}} \right)^b} = {x^{ab}} $ .
 $ = ({6^{\dfrac{1}{3} \times 3}}{x^{9 \times \dfrac{1}{3}}}{y^{18 \times \dfrac{1}{3}}}) $
Common factors from the number and the denominator cancel each other.
 $ = (6{x^3}{y^6}) $
Hence, $ \sqrt[3]{{216{x^9}{y^{18}}}} = (6{x^3}{y^6}) $
This is the required solution.
So, the correct answer is “$(6{x^3}{y^6}) $”.

Note: Know the concepts of squares and cubes. Square is the number multiplied itself and cube it the number multiplied thrice. Square is the product of same number twice such as $ {n^2} = n \times n $ for Example square of $ 2 $ is $ {2^2} = 2 \times 2 $ simplified form of squared number is $ {2^2} = 2 \times 2 = 4 $ . You should be very good in multiples. As, ultimately your answer depends on the multiplication of the numbers. Also, know the basic difference between the cubes and cube roots and apply accordingly. Cube root can be defined as the number which produces a given number when cubed.