Answer:
The solution is:
Part A. [tex]\sqrt{5}^{\frac{7k}{3}})[/tex] which is sqrt(5)^7k/3[/tex]
Part B. k = 18/7
Step-by-step explanation:
Part A.
To solve this part, we're going two use THREE important properties of exponents:
1. [tex](x^{n})^{m} = x^{nm}[/tex]
2. [tex]\frac{x^{n}}{x^{m}} = x^{n-m}[/tex]
3. [tex]\sqrt[n]{x^{m}} = x^{\frac{m}{n} }[/tex]
Let's work the numerator using the properties 1, 2 and 3:
[tex](\sqrt{5}^{3} )^{\frac{k}{9} } } = (\sqrt{5}^{3\frac{k}{9}}) = (\sqrt{5}^{\frac{k}{3}})[/tex]
Let's work the denominator using the properties 1, 2 and 3:
[tex](\sqrt{5}^{6} )^{-\frac{k}{3} } } = (\sqrt{5}^{6\frac{k}{3}}) = (\sqrt{5}^(2k))[/tex]
Now dividing the numerator by the denominator:
[tex]\sqrt{5}^{\frac{k}{3}-(-2k)})=\sqrt{5}^{\frac{7k}{3}})[/tex]
Part B
if [tex]5^{\frac{3}{2} } 5^{\frac{3}{2}} = \sqrt{5}^{\frac{7k}{3}})[/tex]
Then:
[tex]5^{3} = 5^{\frac{7k}{6}})[/tex]
So [tex]\frac{7k}{6}} = 3[/tex]
Solving for k, we have:
k = 18/7
