To solve for the variables \( m \) and \( n \) given the two equations:
[tex]\[ 2^m \times \left(\frac{1}{8}\right)^n = 128 \][/tex]
[tex]\[ 4m \div 2 - 4m = \frac{1}{16} \][/tex]
We'll approach this step by step.
### Step 1: Simplify the Second Equation
Starting with the second equation:
[tex]\[ 4m \div 2 - 4m = \frac{1}{16} \][/tex]
Dividing \( 4m \) by 2 simplifies to \( 2m \):
[tex]\[ 2m - 4m = \frac{1}{16} \][/tex]
Combine like terms:
[tex]\[ -2m = \frac{1}{16} \][/tex]
Solving for \( m \):
[tex]\[ m = -\frac{1}{32} \][/tex]
### Step 2: Substitute \( m \) into the First Equation
Now, we substitute \( m = -\frac{1}{32} \) into the first equation:
[tex]\[ 2^m \times \left(\frac{1}{8}\right)^n = 128 \][/tex]
First, express \( \frac{1}{8} \) as a power of 2:
[tex]\[ \frac{1}{8} = 2^{-3} \][/tex]
The first equation now becomes:
[tex]\[ 2^m \times (2^{-3})^n = 128 \][/tex]
Simplify the exponent on the left side:
[tex]\[ 2^m \times 2^{-3n} = 128 \][/tex]
Combine the exponents:
[tex]\[ 2^{m-3n} = 128 \][/tex]
Now, express 128 as a power of 2:
[tex]\[ 128 = 2^7 \][/tex]
So the equation simplifies to:
[tex]\[ 2^{m - 3n} = 2^7 \][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ m - 3n = 7 \][/tex]
### Step 3: Solve for \( n \) using \( m = -\frac{1}{32} \)
Substitute \( m = -\frac{1}{32} \) into the exponent equation:
[tex]\[ -\frac{1}{32} - 3n = 7 \][/tex]
Add \(\frac{1}{32}\) to both sides:
[tex]\[ -3n = 7 + \frac{1}{32} \][/tex]
To add the fractions, convert 7 to a fraction with the same denominator:
[tex]\[ 7 = \frac{224}{32} \][/tex]
So:
[tex]\[ -3n = \frac{224}{32} + \frac{1}{32} \][/tex]
[tex]\[ -3n = \frac{225}{32} \][/tex]
Divide both sides by -3:
[tex]\[ n = -\frac{225}{32 \times 3} \][/tex]
[tex]\[ n = -\frac{225}{96} \][/tex]
Simplify \(-\frac{225}{96}\):
[tex]\[ n = -\frac{225}{96} = -\frac{75}{32} \][/tex]
### Step 4: Box the Solutions
We have found that:
[tex]\[ m = -\frac{1}{32} \][/tex]
[tex]\[ n = -\frac{75}{32} \][/tex]
So, the solutions are:
[tex]\[ m = -0.03125 \][/tex]
[tex]\[ n = -2.34375 \][/tex]
Comparing this with the given results, we have the following possible solutions:
1. \( m = -0.03125 \), \( n = -2.34375 \)
2. \( m = -0.03125 \), \( n = -2.34375 - 3.0215734278848i \)
3. \( m = -0.03125 \), \( n = -2.34375 + 3.0215734278848i \)
Thus, these are all the solutions to the given equations.
