To solve the given equations for [tex]\( x \)[/tex], we will solve each equation step-by-step.
### Equation 1
The first equation is:
[tex]\[ 13 - 7^x = \left(\sqrt[4]{7}\right)^3 \][/tex]
Step 1: Simplify [tex]\(\left(\sqrt[4]{7}\right)^3\)[/tex].
[tex]\[
\left(\sqrt[4]{7}\right)^3 = 7^{3/4}
\][/tex]
Step 2: Substitute back into the equation.
[tex]\[
13 - 7^x = 7^{3/4}
\][/tex]
Step 3: Isolate [tex]\( 7^x \)[/tex] by subtracting 13 from both sides.
[tex]\[
-7^x = 7^{3/4} - 13
\][/tex]
Thus, moving 7^x to the right:
[tex]\[
7^x = 13 - 7^{3/4}
\][/tex]
Upon solving this equation, we find:
[tex]\[ x = 1.11152032583817 \][/tex]
### Equation 2
The second equation is:
[tex]\[ 14 - \sqrt[5]{5} = 25^x \][/tex]
Step 1: Simplify [tex]\(\sqrt[5]{5}\)[/tex].
[tex]\[
\sqrt[5]{5} = 5^{1/5}
\][/tex]
Step 2: Substitute back into the equation.
[tex]\[
14 - 5^{1/5} = 25^x
\][/tex]
Step 3: Recognize that [tex]\( 25 = 5^2 \)[/tex], so [tex]\( 25^x = (5^2)^x = 5^{2x} \)[/tex].
[tex]\[
14 - 5^{1/5} = 5^{2x}
\][/tex]
To solve for [tex]\( x \)[/tex], we will equate [tex]\( 5^{2x} \)[/tex] to the simplified left side.
Upon solving this equation, we find:
[tex]\[ x = 0.787636558929622 \][/tex]
There is another solution in the complex plane:
[tex]\[ x = 0.787636558929622 + 1.95198126583117i \][/tex]
### Summary
The solutions are:
1. For [tex]\( 13 - 7^x = (\sqrt[4]{7})^3 \)[/tex]:
[tex]\[ x = 1.11152032583817 \][/tex]
2. For [tex]\( 14 - \sqrt[5]{5} = 25^x \)[/tex]:
[tex]\[ x = 0.787636558929622 \][/tex]
and
[tex]\[ x = 0.787636558929622 + 1.95198126583117i \][/tex]
