Sure, let's solve the given expression step-by-step:
The given expression is:
[tex]\[ \sqrt[4]{8 x^7 y^{11}} \times \sqrt[4]{32 x y} \][/tex]
First, we use the property of radicals that states:
[tex]\[ \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b} \][/tex]
So, we can combine the two radicals into one:
[tex]\[ \sqrt[4]{8 x^7 y^{11} \times 32 x y} \][/tex]
Next, let's multiply the terms inside the radical:
1. Coefficients:
[tex]\[ 8 \times 32 = 256 \][/tex]
2. x terms:
[tex]\[ x^7 \times x = x^{7+1} = x^8 \][/tex]
3. y terms:
[tex]\[ y^{11} \times y = y^{11+1} = y^{12} \][/tex]
Putting it all together, we get:
[tex]\[ \sqrt[4]{256 x^8 y^{12}} \][/tex]
Now, we simplify the expression inside the radical:
1. Fourth root of 256:
[tex]\[ \sqrt[4]{256} = 4 \][/tex]
(because [tex]\( 4^4 = 256 \)[/tex])
2. Fourth root of [tex]\( x^8 \)[/tex]:
[tex]\[ \sqrt[4]{x^8} = x^{8/4} = x^2 \][/tex]
3. Fourth root of [tex]\( y^{12} \)[/tex]:
[tex]\[ \sqrt[4]{y^{12}} = y^{12/4} = y^3 \][/tex]
Combine these results to get the final simplified expression:
[tex]\[ 4 x^2 y^3 \][/tex]
Thus, the solution to the given expression is:
[tex]\[ \sqrt[4]{8 x^7 y^{11}} \times \sqrt[4]{32 x y} = 4 x^2 y^3 \][/tex]
