+8 Simplify the following expression into a simplified fraction,
\(\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}\)
Let's start by simplifying the terms inside the square root.
First, we can simplify the denominator of the fraction inside the square root by multiplying both the numerator and denominator by sqrt(5):
(5/sqrt(80) + sqrt(845)/9 + sqrt(45))/sqrt(5) = (5/sqrt(80)sqrt(5) + sqrt(845)/9sqrt(5) + sqrt(45)sqrt(5))/5 = (5/sqrt(165) + sqrt(845)/3sqrt(5) + sqrt(95))/5 = (5/(4sqrt(5)) + sqrt(845)/(3sqrt(5)) + 3)/5 = (1/(4/sqrt(5)) + sqrt(845)/(3sqrt(5)) + 3/1)/5 = (1/(4/sqrt(5)) + sqrt(845)/(3sqrt(5)) + 3(sqrt(5)/5))/5
Next, we can simplify the first term by multiplying the numerator and denominator by sqrt(5)/sqrt(5):
1/(4/sqrt(5)) = sqrt(5)/4
So the expression inside the square root simplifies to:
sqrt((sqrt(5)/4 + sqrt(845)/(3*sqrt(5)) + 3(sqrt(5)/5))/5)
Now, we can combine the terms with a common denominator:
sqrt((sqrt(5) + 4sqrt(845) + 12)/15sqrt(5))
Finally, we can simplify the expression by rationalizing the denominator:
sqrt((sqrt(5) + 4sqrt(845) + 12)/15sqrt(5)) = sqrt((sqrt(5) + 4sqrt(845) + 12)/(15sqrt(5)) * (sqrt(5)/sqrt(5))) = sqrt((5 + 20sqrt(845) + 60)/(75)) = sqrt((1/15) * (5 + 20sqrt(845) + 60)) = sqrt(1/15) * sqrt(85 + 4sqrt(21125)) = sqrt(85 + 4sqrt(21125))/sqrt(15*15) = (sqrt(85) + 10sqrt(47))/15
Therefore, the simplified expression is (sqrt(85) + 10sqrt(47))/15.
