To determine which expression is equivalent to [tex]\(5 \sqrt[6]{2^7 x^7 y}\)[/tex], where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are positive, we will proceed by simplifying the given expression step-by-step.
First, rewrite the given expression for clarity.
The expression is:
[tex]\[5 \sqrt[6]{2^7 x^7 y}\][/tex]
We can break this down into two parts: the coefficient and the remaining expression inside the 6th root.
1. Simplify the Inside Expression:
[tex]\[\sqrt[6]{2^7 x^7 y}\][/tex]
Recall that [tex]\(\sqrt[6]{a^b} = a^{b/6}\)[/tex]. So,
[tex]\[2^7 x^7 y = (2^7 \cdot x^7 \cdot y)\][/tex]
This can be expressed as:
[tex]\[(2^7 \cdot x^7 \cdot y)^{1/6}\][/tex]
2. Apply the Power Rule:
[tex]\[(2^7 \cdot x^7 \cdot y)^{1/6}\][/tex]
Each term inside the root can be raised to the power of [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[= (2^7)^{1/6} \cdot (x^7)^{1/6} \cdot (y)^{1/6}\][/tex]
[tex]\[= 2^{7/6} \cdot x^{7/6} \cdot y^{1/6}\][/tex]
3. Combine with the Coefficient:
The original expression includes a coefficient of 5:
[tex]\[5 \cdot 2^{7/6} \cdot x^{7/6} \cdot y^{1/6}\][/tex]
4. Match with Given Options:
We need to examine the given choices:
Option A:
[tex]\[7 \times (2xy)^{1/6}\][/tex]
[tex]\[= 7 \times 2^{1/6} \cdot x^{1/6} \cdot y^{1/6}\][/tex]
This does not match our original expression.
Option B:
[tex]\[10 x (2xy)^{1/6}\][/tex]
[tex]\[= 10 x \cdot 2^{1/6} \cdot x^{1/6} \cdot y^{1/6}\][/tex]
[tex]\[= 10 \cdot 2^{1/6} \cdot x \cdot x^{1/6} \cdot y^{1/6}\][/tex]
[tex]\[= 10 \cdot 2^{1/6} \cdot x^{1 + 1/6} \cdot y^{1/6}\][/tex]
[tex]\[= 10 \cdot 2^{1/6} \cdot x^{7/6} \cdot y^{1/6}\][/tex]
Notice this matches [tex]\(5 \cdot 2^{7/6} \cdot x^{7/6} \cdot y^{1/6}\)[/tex] because we can split 10 into 25 and compensate for the coefficient difference:
[tex]\[10 = 2 5\][/tex]
This gives:
[tex]\[5 \cdot 2^{1+1/6} \cdot x^{7/6} \cdot y^{1/6}\][/tex]
This simplifies to our expression from the previous step.
Option C:
[tex]\[69 x (2xy)^{1/6}\][/tex]
By following a similar approach as Option B, it's clear that multiplying by 69 is not equivalent to the original 5 coefficient, and it doesn't match our required power expressions inside the root.
Option D:
[tex]\[320 x^6 (2xy)^{1/6}\][/tex]
Multiplying by 320 and adding the different powers makes this option obviously incorrect for matching our simplified expression.
As we see, Option B:
[tex]\[ 10 \times x \times (2 x y)^{\frac{1}{6}} \][/tex]
matches our simplified form.
Thus the correct answer is:
[tex]\[ \boxed{B} \][/tex]
