Question

How do you simplify the following expression?
 $ {(4{x^3}y)^{\dfrac{1}{2}}} \times \sqrt[3] {{8x{y^2}}} $

Answer 1

Hint: First of all we will take the given expression, and then we will use the different laws of power and exponents and also use the concepts of squares and square-root and then simplify for the required value.

Complete step-by-step answer:
Take the given expression –
${(4{x^3}y)^{\dfrac{1}{2}}} \times \sqrt[3] {{8x{y^2}}}$
Convert the given cube-root of the given expression in the form of $\dfrac{1}{3}$and write the power to the term.
$ = {(4{x^3}y)^{\dfrac{1}{2}}} \times {(8x{y^2})^{\dfrac{1}{3}}}$
By using the law of power and exponent which states that when there is power to the power, ${({a^x})^y} = {a^{xy}}$. It is also known as the power of a power rule.
 \[ = ({4^{\dfrac{1}{2}}}{x^{\dfrac{3}{2}}}{y^{\dfrac{1}{2}}}) \times ({8^{\dfrac{1}{3}}}{x^{\dfrac{1}{3}}}{y^{\dfrac{2}{3}}})\]
Using the concept of square and square-roots. Squares are the number when multiplied with itself.
 \[ = ({2^{2{\text{ }} \times }}^{\dfrac{1}{2}}{x^{\dfrac{3}{2}}}{y^{\dfrac{1}{2}}}) \times ({2^{3\, \times }}^{\dfrac{1}{3}}{x^{\dfrac{1}{3}}}{y^{\dfrac{2}{3}}})\]
Same value in the numerator and the denominator cancel each other. Therefore remove from the first part of the above expression and similarly remove from the second part of the above expression.
 \[ = (2{x^{\dfrac{3}{2}}}{y^{\dfrac{1}{2}}}) \times (2{x^{\dfrac{1}{3}}}{y^{\dfrac{2}{3}}})\]
Now, arrange the all like terms together.
 \[ = (2.2.{x^{\dfrac{3}{2}}}.{x^{\dfrac{1}{3}}}.{y^{\dfrac{1}{2}}}.{y^{\dfrac{2}{3}}})\]
Simplify the product of constants in the above equation.
 \[ = (4.{x^{\dfrac{3}{2}}}.{x^{\dfrac{1}{3}}}.{y^{\dfrac{1}{2}}}.{y^{\dfrac{2}{3}}})\]
Again, using the law of power and exponents identity when bases are same powers are added which is also known as the multiplicative rule,
Simplify the above fractions taking the LCM (least common multiple).
 \[ = (4.{x^{\dfrac{{3 + 2}}{6}}}.{y^{\dfrac{{3 + 2}}{6}}})\]
Simplify the above expression –
 \[ = (4.{x^{\dfrac{5}{6}}}.{y^{\dfrac{5}{6}}})\]
This is the required solution.
So, the correct answer is “ \[ (4\times{x^{\dfrac{5}{6}}}\times{y^{\dfrac{5}{6}}})\] ”.

Note: Be careful while using the laws of powers and exponents. Remember one basic rule in the law of power and exponents when the exponents of the numbers are added when the numbers are multiplied while subtracted when the numbers are divided.