To evaluate the expression (cube root of 7 multiplied by the square root of 7) over (the sixth root of 7 to the power of 5), we will break it down step by step. Let’s denote 7 as x for simplicity:
Step 1: Write expressions in exponent form
Cube root of 7: 7^(1/3)
Square root of 7: 7^(1/2)
Sixth root of 7: 7^(1/6)
Step 2: Substitute the expressions
The expression can now be rewritten as:
(x^(1/3) * x^(1/2)) / (x^(1/6))^5
Step 3: Calculate the denominator
Applying the power of a power rule in exponents:
(x^(1/6))^5 = x^(5/6)
Step 4: Rewrite the expression
Now, plug this back into our expression:
(x^(1/3) * x^(1/2)) / x^(5/6)
Step 5: Simplify the numerator
When multiplying like bases, we add the exponents:
x^(1/3 + 1/2) = x^(2/6 + 3/6) = x^(5/6)
Step 6: Substitute back
Now we have:
(x^(5/6)) / (x^(5/6))
Step 7: Cancel out
Now we cancel the common base:
x^(5/6 – 5/6) = x^0 = 1
Conclusion
Thus, the final result for the given expression:
1