Solving the Exponential Equation
The problem requires us to solve for $x$ in the equation $(125)^{1.3} \times (25)^{1.8} \div (5)^{2.5} = 5^x$ and then calculate the value of $18x - 5$. We can simplify the equation by expressing all bases as powers of 5.
Simplifying the Expression
- Express bases as powers of 5:
- $125 = 5^3$
- $25 = 5^2$
- $5 = 5^1$
- Substitute into the equation: $ (5^3)^{1.3} \times (5^2)^{1.8} \div (5^1)^{2.5} = 5^x $
- Apply the power of a power rule $(a^m)^n = a^{m \times n}$: $ 5^{3 \times 1.3} \times 5^{2 \times 1.8} \div 5^{1 \times 2.5} = 5^x $ $ 5^{3.9} \times 5^{3.6} \div 5^{2.5} = 5^x $
- Apply exponent rules for multiplication and division ($a^m \times a^n = a^{m+n}$ and $a^m \div a^n = a^{m-n}$): $ 5^{3.9 + 3.6 - 2.5} = 5^x $ $ 5^{7.5 - 2.5} = 5^x $ $ 5^5 = 5^x $
- Equate the exponents: $ x = 5 $
Calculating the Final Value
Now substitute the value of $x$ into the expression $18x - 5$.
- Substitute $x=5$: $ 18x - 5 = 18(5) - 5 $
- Perform the calculation: $ 90 - 5 = 85 $
The value of $18x - 5$ is 85.
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 \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
