Question
(k) \( \begin{array}{l}2^{x}=0,125 \\ 2^{x}=0,125\end{array} \)
Ask by Schmidt Coleman. in South Africa
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( x = -3 \).
Solution
Solve the system of equations by following steps:
- step0: Solution:
\(\left\{ \begin{array}{l}2^{x}=0.125\\2^{x}=0.125\end{array}\right.\)
- step1: Calculate:
\(\left\{ \begin{array}{l}x=-3\\x=-3\end{array}\right.\)
- step2: Find the intersection:
\(x=-3\)
- step3: Check the solution:
\(x=-3\)
To solve the system of equations:
\[
\begin{array}{l}
2^{x} = 0.125 \\
2^{x} = 0.125
\end{array}
\]
We can see that both equations are identical.
1. **Convert \(0.125\) to a power of \(2\)**:
\[
0.125 = \frac{1}{8} = \frac{1}{2^3} = 2^{-3}
\]
2. **Set the equations equal**:
\[
2^{x} = 2^{-3}
\]
3. **Since the bases are the same, we can equate the exponents**:
\[
x = -3
\]
Thus, the solution to the system of equations is:
\[
x = -3
\]
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Simplify this solution Mind Expander
To solve \( 2^{x} = 0.125 \), first, we can express 0.125 as a power of 2. Notably, \( 0.125 = \frac{1}{8} = 2^{-3} \). Therefore, we can rewrite the equation as \( 2^{x} = 2^{-3} \). By equating the exponents, we get \( x = -3 \). In a real-world context, understanding exponents and their applications is handy, especially in finance with interest rates, or in computing, where binary systems are foundational. It's essential to grasp the concept of exponents to manipulate various equations effectively, whether in algorithms or data storage calculations!
