91Ó°ÊÓ

Understanding how to simplify radicals, such as the fourth root of 81, can be fascinating. First, let's break down the concept of the 'fourth root' to help you understand this better.

The fourth root of a number is a value that, when multiplied by itself four times, gives the original number.
In mathematical notation, the fourth root of 81 is written as \(\sqrt[4]{81}\).

For example, \[ \sqrt[4]{16} = 2 \] because \[ 2 \cdot 2 \cdot 2 \cdot 2 = 16 \].

So, in the case of 81, we need to find a number that, when raised to the power of 4, equals 81. We will explore this more as we dive deeper.

Next, let’s talk about ‘exponents and powers.' Exponents indicate how many times a number, called the base, is multiplied by itself.
For example, in \(3^4\), 3 is the base and 4 is the exponent, meaning \(3 \times 3\times 3 \times 3 = 81\).

So, if we express 81 as a power of another number, we get: \[ 81 = 3^4 \].

This is a crucial step in simplifying the expression \(\sqrt[4]{81}\), where we recognize that 81 is \(3^4\).
With this understanding, it becomes easier to solve the problem.

Finally, let's address the 'simplification of expressions.' Simplifying expressions involves making them as straightforward as possible.
In this exercise, understanding that \(\sqrt[4]{81}\) can be viewed through the lens of exponents helps.
We know from step 2 that \(\sqrt[4]{81}\) is the same as \(\sqrt[4]{3^4}\).

The next step involves applying the properties of roots and exponents.
When you take the fourth root of \(3^4\), it simplifies to the base itself, which is 3.

So, \(\sqrt[4]{3^4} = 3 \).
Thus, simplifying \(\sqrt[4]{81}\) gives us the final answer: 3.

Remember, whenever you see an expression that needs simplifying, break it down into smaller parts.
This makes the process much more manageable and easier to understand.