Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive.
1) 1 over square root of 3 + 5 (it's 1 over the whole thing, so it's 1 over square root of 3 and the + 5 isn't under the radical but it's still in the denominator)
2) square root of 63 + 2 square root of 28 - 5 square root of 7
Please explain and show steps!

Answer 1

\[1/\sqrt{3} + 5\]

Answer 2

@imranmeah91 @ishaan94 @pokemon23 someone help

Answer 3

Ishaan got this.. pro math wizard

Answer 4

\[\frac{\sqrt{3} - 5}{\sqrt{3} - 5}\times \frac{1}{\sqrt{3} + 5}\]
\[\frac{\sqrt{3} - 5}{3 - 5} = \frac{\sqrt{3} - 5}{-2} = \frac{5 - \sqrt{3}}{2}\]

Answer 5

like a boss

Answer 6

lol, like a sir

Answer 7

@trolololcat Can you do the second part on your own now? Or, need I show you second one too?

Answer 8

This is really new to me show me the second one too please

Answer 9

2. square root of 63 + 2 square root of 28 - 5 square root of 7

Solution: \(\sqrt{63} + 2 \sqrt{28} - 5 \sqrt{7} = \sqrt{7 \times 9} + 2 \sqrt{4 \times 7} - 5\sqrt{7} \)
\[\implies\sqrt{7 \times 3^2} + 2 \sqrt{2^2 \times 7} - 5\sqrt{7} = 3\sqrt{7} + 4\sqrt{7} - 5\sqrt{7} \]
\[\implies \sqrt{7} \left(3 + 4 - 5 \right) = 2\sqrt{7} \]

Answer 10

@ishaan94 Do you think you can help me on two more questions? >.<
1)\[\sqrt[3]{2}+1/\sqrt[3]{4}\]
2) 2/1+ square root of 2

Answer 11

I hope you meant to simply the expression, right?

Answer 12

\[1. \:\:\:\:\:\: \ \frac{\sqrt[3]{2} + 1}{\sqrt[3]{2^2}} \implies \large \frac{2^{\frac{1}{3}}}{2^{\frac{2}{3}}} + \frac{1}{\sqrt[3]{2^2}} \implies 2^{\frac{-1}{3}} + \frac{1}{\sqrt[3]{2^2}}\]

Answer 13

yes to simplify and rationalize denominators

Answer 14

I'm confused about 1), what did you do first and what exactly is the answer o_o

Answer 15

argh, I'm sorry... It took time I was on chat :/

Lets do it again, ignore my previous solution...

\[\frac{\sqrt[3]{2} + 1}{\sqrt[3]{2^2}} \times \frac{\sqrt[3]{2}}{\sqrt[3]{2}} = \frac{\sqrt[3]{4} + \sqrt[3]{2}}{2}\]

Answer 16

What I did is got rid of the cube-root of the denominator, that's it, that's all I have done by multiplying and dividing \(\sqrt[3]{2}\).

Answer 17

For the second part.

\[\frac{1}{1 + \sqrt{2}} \times \frac{1-\sqrt{2}}{1 - \sqrt{2}} = \frac{1 - \sqrt{2}}{1 - 2} = \frac{\sqrt{2} -1}{1} = \sqrt{2} - 1\]

Answer 18

so its \[\sqrt[3]{6}/2 ?\]

Answer 19

No, for the first question, I don't think you can simplify it further than my final expression :/

Answer 20

@ishaan94 why can't you combine them?

Answer 21

@Chlorophyll can you double check 1) for us of the second set lol

Answer 22

Um their powers are different, when powers are different you can multiply them to each-other, divide them but can not add or subtract.

Answer 23

@Ishaan94 dont both of them have the power of 3?

Answer 24

argh what I meant was if both the bases and the powers are same, you need both the base and power to be equal to add or subtract.

Answer 25

@Ishaan94 so you think its safe to put that as my final answer

Answer 26

I think so

Answer 27

@amistre64 you think so too?

Answer 28

im reading the original post and seeing something different get answered

Answer 29

what is the original problem?

Answer 30

1) 1 over square root of 3 + 5 (it's 1 over the whole thing, so it's 1 over square root of 3 and the + 5 isn't under the radical but it's still in the denominator)

1/(sqrt(3) + 5)

Answer 31

then yeah, i read the original wrong

Answer 32

a^2-b^2 = (a+b)(a-b)

so if we have somehting of the form (a-b) we multiply it by an opposite operation to rationalize it

Answer 33

amistre, this one ↓ ↓\[\frac{\sqrt[3]{2} + 1}{\sqrt[3]{2^2}} \times \frac{\sqrt[3]{2}}{\sqrt[3]{2}} = \frac{\sqrt[3]{4} + \sqrt[3]{2}}{2}\]
Can you simplify it further? or, Add the terms in Numerator? I don't think so.

Answer 34

you can factor out a cbrt(2) but then you created it to begin so thats not realy a simplification is it

otherwise it loks fine to me

Answer 35

It's probably correct by applying conjugation: 1/ (sqrt3 + 5)
= 1/ (sqrt3 + 5) * (sqrt3 - 5)/ (sqrt3 -5)
= ( sqrt3 -5) / ( 3 - 25)
= -( sqrt3 -5) / 22

Answer 36

My PC's too slow at peak time!

Answer 37

= ( 5- sqrt3 ) / 22
Look much more beautiful :)