Question

How do you simplify $\sqrt[5]{{160{x^{12}}{y^{15}}}}$?

Answer 1

Hint: The power is used to express mathematical equations in the short form; it is an expression that represents the repeated multiplication of the same factor. For example - $2 \times 2 \times 2$ can be expressed as ${2^3}$. Here, the number two is called the base and the exponent represents the number of times the base is used as the factor. Here we will apply the power and exponent rule and simplify the eighth rule of the given expression with the cube of the given term

Complete step by step answer:
Take the given expression: $\sqrt[5]{{160{x^{12}}{y^{15}}}}$
The above expression can be re-written as - $ = {(160{x^{12}}{y^{15}})^{\dfrac{1}{5}}}$
By applying the product of power rule in the above expression, states that when there is power to the power, ${({a^x})^y} = {a^{xy}}$.
So, the expression becomes
$ = {(160)^{\dfrac{1}{5}}}{(x)^{12 \times \dfrac{1}{5}}}{({y^{15}})^{\dfrac{1}{5}}}$
Now, frame the constant term in the form of power and exponent –
$ = {(32 \times 5)^{\dfrac{1}{5}}}{(x)^{12 \times \dfrac{1}{5}}}{({y^{15}})^{\dfrac{1}{5}}}$
Common factors from the numerator and the denominator cancels each other.
$ = {({2^5})^{\dfrac{1}{5}}} \times {(5)^{\dfrac{1}{5}}}{(x)^{\dfrac{{12}}{5}}}({y^3})$
Simplify the above expression –
$ = (2) \times {(5)^{\dfrac{1}{5}}}{(x)^{\dfrac{{12}}{5}}}({y^3})$
This is the required solution.

Note: Don’t be confused between power of a power rule and the power of the product rule.
Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
Product of powers rule
Quotient of powers rule
Power of a power rule
Power of a product rule
Power of a quotient rule
Zero power rule
Negative exponent rule