To solve the equation [tex]\(100^{x+1} \div 10=0.01\)[/tex], we can follow these detailed steps:
1. Simplify the Equation: Start by getting rid of the division by 10.
[tex]\[
\frac{100^{x+1}}{10} = 0.01
\][/tex]
Multiply both sides by 10 to eliminate the fraction:
[tex]\[
100^{x+1} = 0.01 \times 10
\][/tex]
[tex]\[
100^{x+1} = 0.1
\][/tex]
2. Rewrite the Numbers in Exponential Form: Express both sides using base 10.
[tex]\[
100 = 10^2 \quad \text{so} \quad 100^{x+1} = (10^2)^{x+1} = 10^{2(x+1)}
\][/tex]
The equation now is:
[tex]\[
10^{2(x+1)} = 0.1
\][/tex]
Recall that [tex]\(0.1 = 10^{-1}\)[/tex], so we can rewrite:
[tex]\[
10^{2(x+1)} = 10^{-1}
\][/tex]
3. Equate the Exponents: Since the bases (10) are the same, set the exponents equal to each other:
[tex]\[
2(x+1) = -1
\][/tex]
4. Solve the Linear Equation:
[tex]\[
2x + 2 = -1
\][/tex]
Subtract 2 from both sides:
[tex]\[
2x = -3
\][/tex]
Divide by 2:
[tex]\[
x = -1.5
\][/tex]
Thus, the real solution to the equation [tex]\(100^{x+1} \div 10 = 0.01\)[/tex] is:
[tex]\[
x = -1.5
\][/tex]
Additionally, if we allow for complex solutions, we get a complex solution involving an imaginary component:
[tex]\[
x = -1.5 + 1.36437635384184i
\][/tex]
So, including both the real and the complex solutions, the complete set of solutions is:
[tex]\[
x = -1.5 \quad \text{and} \quad x = -1.5 + 1.36437635384184i
\][/tex]
