Understanding the Exponential Problem

We are given a problem involving exponents and variables. We need to find the value of $z$ based on the relationships provided:

• $50^{0.92} = x$
• $50^{0.26}= y$
• $x^z = y^3$

The goal is to use the properties of exponents to solve for $z$.

Step-by-Step Solution

Follow these steps to find the value of $z$:

1. Substitute x and y: Start with the equation $x^z = y^3$. Substitute the given expressions for $x$ and $y$ into this equation.
   Substitute $x = 50^{0.92}$ and $y = 50^{0.26}$: $ (50^{0.92})^z = (50^{0.26})^3 $
2. Apply the Power of a Power Rule: Use the exponent rule $(a^m)^n = a^{m \times n}$ to simplify both sides of the equation.
   • Left side: $(50^{0.92})^z = 50^{0.92 \times z}$
   • Right side: $(50^{0.26})^3 = 50^{0.26 \times 3}$
   The equation now becomes: $ 50^{0.92z} = 50^{0.26 \times 3} $
3. Simplify the Exponent on the Right Side: Calculate the product $0.26 \times 3$. $ 0.26 \times 3 = 0.78 $ So the equation is: $ 50^{0.92z} = 50^{0.78} $
4. Equate the Exponents: Since the bases are the same (both are 50), the exponents must be equal. $ 0.92z = 0.78 $
5. Solve for z: Isolate $z$ by dividing both sides by 0.92. $ z = \frac{0.78}{0.92} $

Final Calculation

Now, perform the division to find the value of $z$:

To calculate this value:

Rounding this value to two decimal places, we get $z$ $\approx$ 0.85.

Therefore, the value of $z$ is close to 0.85.