Question

simplify 5 square root of 7 + 12 square root of 6 - 10 square root of 7 - 5 square root of 6.

Answer 1

i need help please

Answer 2

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:
\[\Large \begin{gathered}
5\sqrt 7 + 12\sqrt 6 - 10\sqrt 7 - 5\sqrt 6 = \hfill \\
\hfill \\
= \sqrt 7 \left( {5 - 10} \right) + \sqrt 6 \left( {12 - 5} \right) = ... \hfill \\
\end{gathered} \]

Answer 3

please complete

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

@Michele_Laino

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

that's right!

Answer 14

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

Answer 15

ok!

Answer 16

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

Answer 17

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

Answer 18

am I right?

Answer 19

@Michele_Laino is that the answer to the question?

Answer 20

no, we have to write another step

Answer 21

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

Answer 22

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

Answer 23

yea thats what i got @Michele_Laino

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

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

Answer 28

its not too difficult

just follow what the question is trying to tell you.

Answer 29

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

Answer 30

there is another way to simplify your radical

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

and we have finished

Answer 35

now you can choose the method which do you prefer