Answer:
n = 7/2
Step-by-step explanation:
We are given the equation:-
[tex] \displaystyle \large{8 \times \sqrt{2} = {2}^{n} }[/tex]
To solve an exponential equation, first, we convert the whole equation with same base.
Let our main base is 2 for whole equation, the following number must be:-
- 8 = 2•2•2 = 2^3
- √2 = 2^(1/2) —> Law of Exponent
From √2 = 2^(1/2) comes from:-
[tex] \displaystyle \large{ {a}^{ \frac{m}{ n } } = \sqrt[n]{ {a}^{m} } }[/tex]
[tex] \displaystyle \large{ {2}^{ \frac{1}{ 2 } } = \sqrt{2} }[/tex]
Rewrite the equation with base 2.
[tex] \displaystyle \large{ {2}^{3} \times {2}^{ \frac{1}{2} } = {2}^{n} }[/tex]
Recall the law of exponent:-
[tex] \displaystyle \large{ {a}^{m} \times {a}^{n} = {a}^{m + n} }[/tex]
Therefore:-
[tex] \displaystyle \large{ {2}^{3 + \frac{1}{2} } = {2}^{n} } \\ \displaystyle \large{ {2}^{ \frac{6}{2} + \frac{1}{2} } = {2}^{n} } \\ \displaystyle \large{ {2}^{ \frac{7}{2} } = {2}^{n} }[/tex]
Compare the exponent and thus:-
n = 7/2.
