## anonymous one year ago simplify 5 square root of 7 + 12 square root of 6 - 10 square root of 7 - 5 square root of 6.

1. anonymous

2. Michele_Laino

hint: we can factor out square root of 7 between the first and the third term, furthermore, we can factor out square root of 6 between the second and fourth term, so we can write the subsequent step: $\Large \begin{gathered} 5\sqrt 7 + 12\sqrt 6 - 10\sqrt 7 - 5\sqrt 6 = \hfill \\ \hfill \\ = \sqrt 7 \left( {5 - 10} \right) + \sqrt 6 \left( {12 - 5} \right) = ... \hfill \\ \end{gathered}$

3. Michele_Laino

4. anonymous

honestly i dont know mate im really confused sorry... @Michele_Laino

5. Michele_Laino

following the rules of algebra of radicals, you can only sum similar radicals, namely radicals which have the same square roots as in your case

6. Michele_Laino

now, $5\sqrt 7 ,\; - 10\sqrt 7$ are similar since they both have square root of 7, right?

7. anonymous

i was trying to work it out and the answer i got was 5 square root of 7 - 7 quare root of 6

8. anonymous

@Michele_Laino

9. Michele_Laino

I got this since 5-10= -5, we have: $\sqrt 7 \left( {5 - 10} \right) = - 5\sqrt 7$

10. Michele_Laino

furthermore 12-5=7, so we can write: $\sqrt 6 \left( {12 - 5} \right)$

11. Michele_Laino

oops.. $\sqrt 6 \left( {12 - 5} \right) = 7\sqrt 6$

12. anonymous

and that will be 7 square root of 6 - 5 square root of 7 @Michele_Laino

13. Michele_Laino

that's right!

14. anonymous

thank you so much can you help me with one more? @Michele_Laino

15. Michele_Laino

ok!

16. anonymous

simplify square root of 5 (10-4 square root of 2) @Michele_Laino

17. Michele_Laino

we have to apply the distributive property of multiplication over addition, so we can write this: $\Large \sqrt 5 \left( {10 - 4\sqrt 2 } \right) = 10\sqrt 5 - 4\sqrt 5 \sqrt 2$

18. Michele_Laino

am I right?

19. anonymous

@Michele_Laino is that the answer to the question?

20. Michele_Laino

no, we have to write another step

21. anonymous

ohhhhhh okay well as i simplify it i got 5 square root of 2 - 4 square root of 10 @Michele_Laino

22. Michele_Laino

more precisely we can write this: $\sqrt 5 \sqrt 2 = \sqrt {5 \times 2} = \sqrt {10}$

23. anonymous

yea thats what i got @Michele_Laino

24. Michele_Laino

so we get: $\sqrt 5 \left( {10 - 4\sqrt 2 } \right) = 10\sqrt 5 - 4\sqrt {10}$

25. Michele_Laino

furthermore we can note that: $10 = \sqrt {10} \sqrt {10}$

26. Michele_Laino

so we can write: $\sqrt 5 \left( {10 - 4\sqrt 2 } \right) = 10\sqrt 5 - 4\sqrt {10} = \sqrt {10} \sqrt {10} \sqrt 5 - 4\sqrt {10}$

27. Michele_Laino

finally, I factor out sqrt(10) and I get: $\begin{gathered} \sqrt 5 \left( {10 - 4\sqrt 2 } \right) = 10\sqrt 5 - 4\sqrt {10} = \sqrt {10} \sqrt {10} \sqrt 5 - 4\sqrt {10} \hfill \\ = \sqrt {10} \left( {\sqrt {10} \sqrt 5 - 4} \right) \hfill \\ \end{gathered}$

28. anonymous

its not too difficult just follow what the question is trying to tell you.

29. Michele_Laino

recalling taht: $\sqrt {10} \sqrt 5 = \sqrt {10 \times 5} = \sqrt {50}$ we have: $\begin{gathered} \sqrt 5 \left( {10 - 4\sqrt 2 } \right) = 10\sqrt 5 - 4\sqrt {10} = \sqrt {10} \sqrt {10} \sqrt 5 - 4\sqrt {10} \hfill \\ \hfill \\ = \sqrt {10} \left( {\sqrt {10} \sqrt 5 - 4} \right) = \sqrt {10} \left( {\sqrt {50} - 4} \right) \hfill \\ \end{gathered}$

30. Michele_Laino

31. Michele_Laino

I start from the initial expression: $\Large \sqrt 5 \left( {10 - 4\sqrt 2 } \right)$

32. Michele_Laino

and I factor out a 2 inside the parentheses: $\Large 10 - 4\sqrt 2 = 2\left( {5 - 2\sqrt 2 } \right)$

33. Michele_Laino

so I can write: $\Large \sqrt 5 \left( {10 - 4\sqrt 2 } \right) = \sqrt 5 \cdot 2\left( {5 - 2\sqrt 2 } \right)$

34. Michele_Laino

and we have finished

35. Michele_Laino

now you can choose the method which do you prefer