91Ó°ÊÓ

Fractional exponents can seem tricky at first, but they are much easier to understand with a little practice.
When dealing with fractional exponents, the numerator and the denominator each have a specific role:
- The numerator (top number) of the fraction tells you the power to which the base is raised.
- The denominator (bottom number) tells you the root to be taken.
For the example given, \(y^{\frac{2}{3}}\), the exponent is \(\frac{2}{3}\).
This can be broken down into:
- Raising \(y\) to the power of 2
- Taking the cube root of that result
You can work in different orders, but both yield the same result.

Radicals are used in math to indicate roots. The most familiar radical is the square root, but there are other types as well.
The general form for a radical is \[\sqrt[n]{a}\] where 'a' is the radicand (the number under the radical sign), and 'n' is the index (which root it is).
So, \ \sqrt[3]{y^2} \ indicates the cube root of \(y^2\).
In the step-by-step solution, we converted the fractional exponent \(\frac{2}{3}\) into a radical form:
\ (y \^ 2)^\frac{1}{3} \ which simplifies to \ \sqrt[3]{y^2}\.
The key is to understand that radicals and fractional exponents are two ways of expressing the same thing.

Algebra is a fundamental part of mathematics dealing with symbols and the rules for manipulating those symbols.
It's about finding the unknowns, often referred to as variables, like \(y\) in our example.
When converting between fractional exponents and radicals, you'll often need to manipulate the expressions algebraically.
This may include:
- Identifying the base and exponent
- Breaking down the exponent into its fractional parts
Understanding these basics allows you to solve more complex expressions step-by-step, systematically.
In the given problem, recognizing that \(\frac{2}{3}\) is our exponent is crucial.
By breaking it down, we can see that it's telling us to do two things: square \(y\) and then take the cube root.
This systematic approach, foundational in algebra, leads us to the correct radical notation: \ \sqrt[3]{y^2} \