To solve the equation [tex]\(\left(2^{\frac{x}{5}}\right)\left(2^{\frac{x}{2}}\right) = 2^{14}\)[/tex], we can proceed step by step as follows:
1. Combine the Exponents Using Properties of Exponents:
The given equation is:
[tex]\[
\left(2^{\frac{x}{5}}\right)\left(2^{\frac{x}{2}}\right) = 2^{14}
\][/tex]
We can use the property of exponents which states that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. This allows us to combine the exponents on the left-hand side:
[tex]\[
2^{\frac{x}{5} + \frac{x}{2}} = 2^{14}
\][/tex]
2. Setting the Exponents Equal:
Since the bases are the same (both are 2), we can set the exponents equal to each other. This gives us the equation:
[tex]\[
\frac{x}{5} + \frac{x}{2} = 14
\][/tex]
3. Solving the Equation for [tex]\(x\)[/tex]:
To solve the equation for [tex]\(x\)[/tex], we need to combine the fractions on the left-hand side. To do this, find a common denominator. The denominators are 5 and 2, so the common denominator is 10.
Rewriting each fraction with a denominator of 10:
[tex]\[
\frac{x}{5} = \frac{2x}{10}
\][/tex]
[tex]\[
\frac{x}{2} = \frac{5x}{10}
\][/tex]
Now add these fractions:
[tex]\[
\frac{2x}{10} + \frac{5x}{10} = \frac{2x + 5x}{10} = \frac{7x}{10}
\][/tex]
So, the equation becomes:
[tex]\[
\frac{7x}{10} = 14
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we need to isolate it by performing the following steps:
[tex]\[
7x = 14 \cdot 10
\][/tex]
[tex]\[
7x = 140
\][/tex]
[tex]\[
x = \frac{140}{7}
\][/tex]
[tex]\[
x = 20
\][/tex]
So, the solution to the equation [tex]\(\left(2^{\frac{x}{5}}\right)\left(2^{\frac{x}{2}}\right) = 2^{14}\)[/tex] is:
[tex]\[
x = 20
\][/tex]
