Exponent Calculation: Solving for 'z'

This problem involves solving an equation with exponents. We are given three pieces of information:

• The first equation relates $x$ to a power of 112: $112^{0.57} = x$
• The second equation relates $y$ to a power of 112: $112^{0.8} = y$
• The third equation links $x$, $y$, and $z$: $x^z = y^6$

Our goal is to find the value of $z$.

Step-by-Step Solution

1. Substitute 'x' and 'y': We can substitute the expressions for $x$ and $y$ from the first two equations into the third equation ($x^z = y^6$).

   This gives us: $(112^{0.57})^z = (112^{0.8})^6$

2. Apply the Power of a Power Rule: When raising a power to another power, we multiply the exponents. The rule is $(a^m)^n = a^{m \times n}$. Applying this to both sides of our equation:

   • Left side: $(112^{0.57})^z = 112^{(0.57 \times z)}$
   • Right side: $(112^{0.8})^6 = 112^{(0.8 \times 6)}$

   So the equation becomes: $112^{(0.57 \times z)} = 112^{(0.8 \times 6)}$

3. Equate the Exponents: Since the bases are the same (112 on both sides of the equation), the exponents must be equal.

   Therefore: $0.57 \times z = 0.8 \times 6$

4. Simplify the Right Side: First, calculate the product on the right side.

   $0.8 \times 6 = 4.8$

   The equation is now: $0.57 \times z = 4.8$

5. Solve for 'z': To find $z$, divide both sides by 0.57.

   $z = \frac{4.8}{0.57}$

6. Calculate the Final Value: Performing the division gives us:

   $z \approx 8.42105...$

Conclusion

The calculated value of $z$ is approximately 8.42105. Comparing this to the given options, the value closest to our result is 8.42.