The problem asks us to find the value of $z$ given the equations:
- $76^{0.44} = x$
- $76^{0.93} = y$
- $x^z = y^3$
Solving for the Exponent z
We can substitute the expressions for $x$ and $y$ into the third equation:
$(76^{0.44})^z = (76^{0.93})^3$
Applying Exponent Rules
Using the rule $(a^m)^n = a^{m \times n}$, we simplify both sides of the equation:
$76^{(0.44 \times z)} = 76^{(0.93 \times 3)}$
Equating Exponents
Since the bases are the same (76), we can equate the exponents:
$0.44 \times z = 0.93 \times 3$
Calculating the Value
First, calculate the right side:
$0.93 \times 3 = 2.79$
Now, the equation is:
$0.44 \times z = 2.79$
Solve for $z$ by dividing both sides by 0.44:
$z = \frac{2.79}{0.44}$
$z \approx 6.3409...$
Final Answer Determination
Comparing the calculated value of $z$ with the given options, the closest value is 6.34.
Therefore, the value of $z$ is close to 6.34.
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 \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
