To simplify the expression [tex]\(\sqrt[4]{4}\)[/tex] by expressing it with radicals, let's follow these steps in a detailed manner:
1. Identify the expression: We need to work with the expression [tex]\(\sqrt[4]{4}\)[/tex], which represents the fourth root of 4.
2. Understand the concept of the fourth root: The fourth root of a number [tex]\(x\)[/tex] is a value [tex]\(y\)[/tex] such that [tex]\(y^4 = x\)[/tex]. In this case, we are looking for a number [tex]\(y\)[/tex] such that [tex]\(y^4 = 4\)[/tex].
3. Express 4 as a power of 2: Notice that 4 can be written as [tex]\(2^2\)[/tex]. That is, [tex]\(4 = 2^2\)[/tex].
4. Apply the fourth root to the expression [tex]\(2^2\)[/tex]: We can rewrite the original expression using the equivalent form of 4:
[tex]\[ \sqrt[4]{4} = \sqrt[4]{2^2} \][/tex]
5. Simplify the root expression: To simplify [tex]\(\sqrt[4]{2^2}\)[/tex], we use the property of radicals [tex]\( \sqrt[n]{a^m} = a^{m/n} \)[/tex]:
[tex]\[ \sqrt[4]{2^2} = 2^{2/4} \][/tex]
6. Simplify the exponent: We simplify the fraction in the exponent:
[tex]\[ 2^{2/4} = 2^{1/2} \][/tex]
7. Convert to radical form: The expression [tex]\(2^{1/2}\)[/tex] is equivalent to the square root of 2:
[tex]\[ 2^{1/2} = \sqrt{2} \][/tex]
Thus, the simplified form of [tex]\(\sqrt[4]{4}\)[/tex] expressed with radicals is:
[tex]\[ \sqrt[4]{4} = \sqrt{2} \][/tex]
Therefore, [tex]\(\sqrt[4]{4}\)[/tex] simplifies to [tex]\(\sqrt{2}\)[/tex].
