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Fourth roots are a way of simplifying expressions with exponents. When we take the fourth root of a number, we are essentially asking: which number, when multiplied by itself four times, equals the given number?
For example, if we have \sqrt[4]{16}\, the number 2 is the fourth root, because \(2 \times 2 \times 2 \times 2 = 16\). This means the fourth root operation is all about finding groups of four identical factors that result in the original number. In our problem, we are looking at \sqrt[4]{48}\, which contains these groups once we perform a prime factorization.
Not all numbers have obvious fourth roots, which is why using prime factorization is a helpful step when simplifying these kinds of expressions.

Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers. A prime number is one that can't be divided by any number except for 1 and itself.
To find the prime factors of a number like 48, you would start by dividing it by the smallest prime, which is 2:

• 48 divided by 2 equals 24.
• Then, 24 divided by 2 equals 12.
• Next, 12 divided by 2 equals 6.
• Finally, 6 divided by 2 equals 3.

This leaves us with the prime factors: \(2 \times 2 \times 2 \times 2 \times 3 = 2^4 \cdot 3\).
By breaking down numbers this way, we can easily identify when we have enough of a number to form a group for a fourth root, such as the four 2's here.

When we talk about perfect fourth powers, we are referring to numbers that can be expressed as another integer raised to the power of four.
For instance, the prime factorization of 48 gave us \(2^4 \cdot 3\). The \(2^4\) here is a perfect fourth power, meaning its fourth root is a whole number, which is 2.
After finding groups of four in the prime factorization, we can simplify the expression by taking the fourth root of those perfect fourth powers. Any leftover factors that do not complete a group remain under the radical.
In our example, since the fourth root of \(2^4\) is 2 and there is no group of four for the factor of 3, we are left with \(2\sqrt[4]{3}\) as our simplified expression. This shows how identifying perfect fourth powers helps in the simplification process.