To solve the expression [tex]\(\sqrt[4]{256 x^4 y^8}\)[/tex], we'll evaluate each component step-by-step. The fourth root of a product is equal to the product of the fourth roots of each factor.
Step 1: Evaluate the fourth root of the constant 256.
[tex]\[ \sqrt[4]{256} \][/tex]
256 is a power of 2:
[tex]\[ 256 = 2^8 \][/tex]
So:
[tex]\[ \sqrt[4]{256} = \sqrt[4]{2^8} \][/tex]
To find the fourth root, we divide the exponent by 4:
[tex]\[ \sqrt[4]{2^8} = 2^{8/4} = 2^2 = 4 \][/tex]
Step 2: Evaluate the fourth root of [tex]\(x^4\)[/tex].
[tex]\[ \sqrt[4]{x^4} \][/tex]
Similarly, we divide the exponent by 4:
[tex]\[ \sqrt[4]{x^4} = (x^4)^{1/4} = x^{4/4} = x^1 = x \][/tex]
Step 3: Evaluate the fourth root of [tex]\(y^8\)[/tex].
[tex]\[ \sqrt[4]{y^8} \][/tex]
Again, we divide the exponent by 4:
[tex]\[ \sqrt[4]{y^8} = (y^8)^{1/4} = y^{8/4} = y^2 \][/tex]
Step 4: Combine the results from Step 1, Step 2, and Step 3:
[tex]\[ \sqrt[4]{256 x^4 y^8} = \sqrt[4]{256} \cdot \sqrt[4]{x^4} \cdot \sqrt[4]{y^8} = 4 \cdot x \cdot y^2 \][/tex]
Thus, the simplified expression is:
[tex]\[ \sqrt[4]{256 x^4 y^8} = 4xy^2 \][/tex]
