Question - Finding the Value of n in an Exponential Equation
Solution:
Given the equation:
(xn)3 = \frac{x18}{x-6}
We can simplify the right side:
\frac{x18}{x-6} = x18 - (-6) = x18 + 6 = x24
Now we have:
(xn)3 = x24
Using the power of a power rule, the left side becomes:
x3n = x24
Since the bases are the same, we can set the exponents equal to each other:
3n = 24
Now solve for n:
n = \frac{24}{3} = 8
The value of n is 8.
Tutor Solution:
To solve the equation \((x^n)^3 = \frac{x^{18}}{x^{-6}}\), we want to find the value of \(n\). Let’s break it down step by step.
**Step 1: Simplify the Right-Hand Side (RHS)**
The right-hand side is \(\frac{x^{18}}{x^{-6}}\). We can simplify this using the laws of exponents. When you divide powers with the same base, you subtract the exponents:
\[ \frac{x^{a}}{x^{b}} = x^{a-b} \]Applying this to our expression:
\[ \frac{x^{18}}{x^{-6}} = x^{18 - (-6)} = x^{18 + 6} = x^{24} \]**Step 2: Rewrite the Equation**
Now that we have simplified the RHS, we rewrite our original equation:
\[ (x^n)^3 = x^{24} \]**Step 3: Apply the Power of a Power Rule**
On the left-hand side (LHS), we can use another exponent rule. The power of a power states:
\[ (x^m)^p = x^{mp} \]So we can rewrite the LHS:
\[ x^{n \cdot 3} = x^{3n} \]**Step 4: Set the Exponents Equal**
Since we now have both sides with the base \(x\), we can set the exponents equal to each other:
\[ 3n = 24 \]**Step 5: Solve for \(n\)**
To find \(n\), we just need to divide both sides by 3:
\[ n = \frac{24}{3} = 8 \]**Conclusion**
The value of \(n\) is \(8\). We have followed the process of simplifying the equation, applying exponent rules, and ultimately isolating the variable. This is crucial for solving exponential equations efficiently.
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