Answer:
[tex]\displaystyle a:b=\frac{1}{3}[/tex]
Step-by-step explanation:
Ratios
We are given the following relations:
[tex]a=\sqrt{7}+\sqrt{c}\qquad \qquad[1][/tex]
[tex]b=\sqrt{63}+\sqrt{d}\qquad \qquad[2][/tex]
[tex]\displaystyle \frac{c}{d}=\frac{1}{9} \qquad \qquad [3][/tex]
From [3]:
[tex]9c=d[/tex]
Replacing into [2]:
[tex]b=\sqrt{63}+\sqrt{9c}[/tex]
We can express 63=9*7:
[tex]b=\sqrt{9*7}+\sqrt{9c}[/tex]
Taking the square root of 9:
[tex]b=3\sqrt{7}+3\sqrt{c}[/tex]
Factoring:
[tex]b=3(\sqrt{7}+\sqrt{c})[/tex]
Find the ration a:b:
[tex]\displaystyle a:b=\frac{\sqrt{7}+\sqrt{c}}{3(\sqrt{7}+\sqrt{c})}[/tex]
Simplifying:
[tex]\boxed{a:b=\frac{1}{3}}[/tex]
