Exponential Values Given

We are given three pieces of information:

• $x = 81^{0.65}$
• $y = 81^{0.26}$
• $x^z = y^7$

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

Solving for z: Step-by-Step

We can use the given information to solve for $z$ by substituting the expressions for $x$ and $y$ into the third equation.

Substituting x and y

Replace $x$ and $y$ in the equation $x^z = y^7$ with their given exponential forms:

$(81^{0.65})^z = (81^{0.26})^7$

Applying Exponent Rules

We use the power of a power rule, which states that $(a^m)^n = a^{m \times n}$. Applying this rule to both sides of the equation:

• Left side: $(81^{0.65})^z = 81^{0.65 \times z}$
• Right side: $(81^{0.26})^7 = 81^{0.26 \times 7}$

So the equation becomes:

$81^{0.65 \times z} = 81^{0.26 \times 7}$

Equating Exponents

Since the bases on both sides of the equation are the same (base 81), the exponents must be equal:

$0.65 \times z = 0.26 \times 7$

Calculating the Value of z

First, calculate the product on the right side:

$0.26 \times 7 = 1.82$

Now the equation is:

$0.65 \times z = 1.82$

To solve for $z$, divide both sides by 0.65:

$z = \frac{1.82}{0.65}$

To simplify the division, we can multiply the numerator and denominator by 100 to remove the decimals:

$z = \frac{182}{65}$

Now, we can perform the division:

$z = 2.8$

Final Answer Check

The calculated value of $z$ is 2.8. This value is listed as one of the options.