Identify components of expression: Identify the components of the expression 1/(1+e^-7). We have a fraction where the numerator is 1 and the denominator is the sum of 1 and e raised to the power of -7.Apply negative exponent rule: Apply the negative exponent rule to e^-7. According to the negative exponent rule, a^-m = 1/a^m. Therefore, e^-7 is equivalent to 1/e^7.Substitute with positive exponent: Substitute e^-7 with its equivalent positive exponent form in the expression. The expression becomes 1/(1 + 1/e^7).Find common denominator: Find a common denominator to combine the terms in the denominator. The common denominator for 1 and 1/e^7 is e^7. We rewrite 1 as e^7/e^7.Rewrite using common denominator: Rewrite the denominator using the common denominator. The expression becomes 1/((e^7/e^7) + (1/e^7)) which simplifies to 1/((e^7 + 1)/e^7).Simplify complex fraction: Simplify the complex fraction. The expression simplifies to 1 * (e^7/(e^7 + 1)) which is e^7/(e^7 + 1).Check for errors: Check for any mathematical errors in the simplification process. No mathematical errors were made in the previous steps.

Grade 8

Solve equations with variable exponents

Full solution

\frac{1}{1+e^{-7}}

Identify components of expression: Identify the components of the expression $\frac{1}{1+e^{-7}}$ . We have a fraction where the numerator is $1$ and the denominator is the sum of $1$ and $e$ raised to the power of $-7$ .
Apply negative exponent rule: Apply the negative exponent rule to $e^{-7}$ . According to the negative exponent rule, $a^{-m} = \frac{1}{a^m}$ . Therefore, $e^{-7}$ is equivalent to $\frac{1}{e^7}$ .
Substitute with positive exponent: Substitute $e^{-7}$ with its equivalent positive exponent form in the expression. $\newline$ The expression becomes $1/(1 + 1/e^7)$ .
Find common denominator: Find a common denominator to combine the terms in the denominator. $\newline$ The common denominator for $1$ and $1/e^7$ is $e^7$ . We rewrite $1$ as $e^7/e^7$ .
Rewrite using common denominator: Rewrite the denominator using the common denominator. $\newline$ The expression becomes $\frac{1}{\left(\frac{e^7}{e^7} + \frac{1}{e^7}\right)}$ which simplifies to $\frac{1}{\left(\frac{e^7 + 1}{e^7}\right)}$ .
Simplify complex fraction: Simplify the complex fraction. $\newline$ The expression simplifies to $1 \times \left(\frac{e^7}{e^7 + 1}\right)$ which is $\frac{e^7}{e^7 + 1}$ .
Check for errors: Check for any mathematical errors in the simplification process. $\newline$ No mathematical errors were made in the previous steps.