Initial Equations Analysis
We are given the following information:
- $7^{0.07} = x$
- $7^{0.07} = y$
- $x^z = y^3$
The goal is to determine the value of $z$.
Deriving the Value of z
From the first two equations, it's clear that both $x$ and $y$ are equal to the same value, $7^{0.07}$. Therefore, we can conclude that $x = y$.
Now, we substitute $x = y$ into the third equation, $x^z = y^3$:
$x^z = x^3$
Since $x = 7^{0.07}$, the base $x$ is a positive number not equal to 1. For the equation $x^z = x^3$ to hold true, the exponents must be equal.
$z = 3$
Conclusion
The exact value of $z$ is 3. This matches Option 3 directly.
Similar Questions
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Important Questions from Surds and Indices
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
The expression \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) is equal to:
Let \(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\) . Then √x is equal to:
If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
What is the value of \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
