Solution:

To simplify the given expression, we will break it down step by step.

Step 1: Evaluate the expression inside the first square root.
- The expression inside the first square root is log 7 base 5.
- This can be rewritten as log 7 / log 5 since log 7 base 5 is equivalent to log 7 divided by log 5.
- Using logarithmic properties, we can evaluate this expression as log 7 / log 5 = log base 5 (7).

Step 2: Evaluate the expression inside the second square root.
- The expression inside the second square root is log 5 base 7.
- This can be rewritten as log 5 / log 7 since log 5 base 7 is equivalent to log 5 divided by log 7.
- Using logarithmic properties, we can evaluate this expression as log 5 / log 7 = log base 7 (5).

Step 3: Substitute the values back into the original expression.
- Now that we have evaluated the expressions inside the square roots, we can substitute them back into the original expression:
- 5^(√(log 7 base 5)) / 7^(√(log 5 base 7))
- This becomes 5^(√(log base 5 (7))) / 7^(√(log base 7 (5)))

Step 4: Simplify the expression further.
- Since the expressions inside the square roots are both logarithmic values, we can simplify them using the property:
√(log base a (b)) = b^(1/log base a (b)).
- Applying this property to our expression, we get:
- 5^(1/log base 5 (7)) / 7^(1/log base 7 (5))

Step 5: Evaluate the logarithmic values.
- We can evaluate the logarithmic values using the change of base formula:
log base a (b) = log base c (b) / log base c (a).
- Applying this formula to our expression, we get:
- 5^(1/(log 7 / log 5)) / 7^(1/(log 5 / log 7))
- Simplifying this further, we have:
- 5^(log 5 / log 7) / 7^(log 7 / log 5)

Step 6: Simplify the expression even more.
- We can simplify the exponents using the property:
(a^b) / (c^b) = (a/c)^b.
- Applying this property to our expression, we get:
- (5^(log 5) / 7^(log 7)) / (5^(log 7) / 7^(log 5))
- Simplifying this further, we have:
- (5 / 7)^(log 5 / log 7) / (7 / 5)^(log 7 / log 5)

Step 7: Simplify the expression using the property of exponents.
- When the bases of the exponents are reciprocals, raising them to the same power will result in 1.
- Applying this property to our expression, we get:
- (5 / 7)

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