How do you simplify 10 over the square root of 2 over 2?

Question

Nathan Axel · Accepted Answer

See a solution process below:

If "the square root of 2 over 2" is #sqrt(2/2)#, the we can write this word problem as: #10/sqrt(2/2)# First, simplify the radical as: #10/sqrt(1) => 10/1 => 10# If "the square root of 2 over 2" is #sqrt(2)/2#, the we can write this word problem as: #10/(sqrt(2)/2) => (10/1)/(sqrt(2)/2)# We can then use this rule for dividing fractions to simplify the expression: #(color(red)(a)/color(blue)(b))/(color(green)(c)/color(purple)(d)) = (color(red)(a) xx color(purple)(d))/(color(blue)(b) xx color(green)(c))# #(color(red)(10)/color(blue)(1))/(color(green)(sqrt(2))/color(purple)(2)) = (color(red)(10) xx color(purple)(2))/(color(blue)(1) xx color(green)(sqrt(2))) => 20/sqrt(2)#

Genesis Allen · Answer

undefined To simplify $ \frac{10}{\sqrt{\frac{2}{2}}} $, you can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of $ \frac{2}{2} $ is $ \frac{2}{2} $.

$$ \frac{10}{\sqrt{\frac{2}{2}}} \times \frac{\sqrt{\frac{2}{2}}}{\sqrt{\frac{2}{2}}} = \frac{10 \cdot \sqrt{\frac{2}{2}}}{\left(\sqrt{\frac{2}{2}}\right)^2} = \frac{10 \cdot \sqrt{\frac{2}{2}}}{\frac{2}{2}} $$

$$ = \frac{10 \cdot \sqrt{\frac{2}{2}}}{1} = 10 \cdot \sqrt{\frac{2}{2}} = 10 \cdot \sqrt{1} = 10 $$

So, $ \frac{10}{\sqrt{\frac{2}{2}}} $ simplifies to $ 10 $.