Solving for z in Exponential Equations

This problem requires us to find the value of an unknown exponent, z, using the properties of exponents and given algebraic relationships.

Given Information

We start with the following definitions:

• $ x = 58^{0.5} $
• $ y = 58^{0.79} $

And we are given the equation relating x and y:

• $ x^z = y^{10} $

Substituting Known Values

Our main goal is to determine the value of z. We can substitute the expressions for x and y from the first two equations into the third equation:

Replacing x with $ 58^{0.5} $ and y with $ 58^{0.79} $ in the equation $ x^z = y^{10} $, we get:

$ (58^{0.5})^z = (58^{0.79})^{10} $

Applying Properties of Exponents

To simplify this equation, we use a fundamental rule of exponents: the power rule, which states that $ (a^m)^n = a^{m \times n} $. We apply this rule to both sides of our equation:

• On the left side: $ (58^{0.5})^z $ becomes $ 58^{(0.5 \times z)} $.
• On the right side: $ (58^{0.79})^{10} $ becomes $ 58^{(0.79 \times 10)} $.

After applying the power rule, the equation is now:

$ 58^{(0.5 \times z)} = 58^{(0.79 \times 10)} $

Equating Exponents

A key principle in algebra is that if two exponential expressions have the same base and are equal, their exponents must also be equal. In our equation, the base is 58 on both sides.

Therefore, we can set the exponents equal to each other:

$ 0.5 \times z = 0.79 \times 10 $

Calculating the Value of z

Let's first compute the product on the right side of the equation:

$ 0.79 \times 10 = 7.9 $

Substituting this back into our equation gives:

$ 0.5 \times z = 7.9 $

To isolate and solve for z, we divide both sides of the equation by 0.5:

$ z = \frac{7.9}{0.5} $

Performing the division calculation:

$ z = 15.8 $

Final Result

The calculation shows that the value of z is 15.8.