Question

How do you solve $x$ to the sixth power over $y$ to the fourth power by simple radicals?

Answer 1

Hint: In this question, we need to first write the given mathematical statement into a mathematical expression. Then, we have to convert the algebraic expression into a simple radical form so as to get to our required answer. We will use a law of radicals, to write the given term into the simplest form of exponent.

Complete step by step solution:
Here, we need to write to convert the given statement into mathematical expression.
So, we are given that “$x$ to the sixth power over $y$ to the fourth power”.
First, we break down the information given to us to try and identify what algebraic expression must look like. We should always know what you are dealing with.
So, in other words, we are given a fraction with the sixth power of $x$ over the fourth power of $y$.
Hence, we have, $\dfrac{{{x^6}}}{{{y^4}}}$.
Now, we have to express this expression in the simplest radical form possible.
Expressing in simplest radical form is nothing but simplifying the radical into the simplest form with no more square roots, cube roots, etc and converting the fractions into a product with the use of negative powers.
So, we know that $\dfrac{1}{{{x^n}}} = {x^{ - n}}$. Hence, we get,
$ \Rightarrow \dfrac{{{x^6}}}{{{y^4}}} = {x^6}{y^{ - 4}}$
Hence, the required answer for x to the sixth power over y to the fourth power in simplest radical form is ${x^6}{y^{ - 4}}$.

Note:
In this question it is important to note here that we used a law of the radical to solve this form i.e., a radical represents a fractional exponent in which the numerator of the fractional exponent is the power of the base and the denominator of the fractional exponent is the index of the radical.