Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 0 \)
Solution
To solve the equation:
\[
6 \cdot 10^{x} + 10^{x + 2} - 106 = 0
\]
follow these steps:
1. **Simplify the Equation:**
Notice that \(10^{x + 2}\) can be rewritten using the properties of exponents:
\[
10^{x + 2} = 10^{x} \cdot 10^{2} = 100 \cdot 10^{x}
\]
Substitute this back into the original equation:
\[
6 \cdot 10^{x} + 100 \cdot 10^{x} - 106 = 0
\]
2. **Combine Like Terms:**
Combine the terms involving \(10^{x}\):
\[
(6 + 100) \cdot 10^{x} - 106 = 0
\]
\[
106 \cdot 10^{x} - 106 = 0
\]
3. **Isolate the Exponential Term:**
Add 106 to both sides:
\[
106 \cdot 10^{x} = 106
\]
Divide both sides by 106:
\[
10^{x} = 1
\]
4. **Solve for \(x\):**
Recall that \(10^{0} = 1\). Therefore:
\[
x = 0
\]
5. **Verify the Solution:**
Substitute \(x = 0\) back into the original equation to ensure it's correct:
\[
6 \cdot 10^{0} + 10^{0 + 2} - 106 = 6 \cdot 1 + 100 - 106 = 6 + 100 - 106 = 0
\]
The equation holds true, so \(x = 0\) is indeed the solution.
**Final Answer:**
\( x = 0 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
Did you know that exponential equations like this one often pop up in real-world situations, such as calculating population growth or in finance when determining compound interest? They might look complicated at first, but they can be solved using logarithmic properties or by graphing to find the intersection points! When solving exponential equations, it's easy to make mistakes, such as neglecting to align the bases or forgetting to apply logarithms correctly. A common pitfall is assuming the equation can be solved simply by manipulating it without considering the characteristics of the functions involved. Remember to check for possible simplifications or factorizations first!
