Simplifying Radicals

[Pencil]

Hi,

This is my first post and I was hoping I can get some help regarding simplifying a radical. I'm trying to practice for a test. I've seen several YouTube videos and they were very helpful but the one I posted below just looks different to the ones I'm accustom to so I don't know how to simplify it any further. I'm not looking for just the answer. I would like a step by step explanation of how you simplified it. Below is a list of the original problem with the multiple choice answers and my failed approach at simplifying it. Thank you for your time and patience!

Problem:

^5(^15+^5)

Multiple Choice Answer:

A) 5^3 + ^5
B) 25 ^3
C) 3^5 + 5
D) 5^3 + 5

My Approach:

^5(^15+^5)
= ^5 ^20
= ^5 ^4 ^5
= ^5 2 ^5

Clearly my approach was not part of the multiple choice answer so any help would be appreciated. Thanks again.

 

[HallsofIvy]

I have no idea what "^5(^15+^5)" means! We often use "a^b" to mean a^b but I have never seen "^" in front of a number before. Do you mean 5^(15+ 20)= 5^{15+ 20}?

 

[lookagain]

I figured out what it means.

Problem:

^5(^15+^5) \(\displaystyle \ \ \ \ \ \ \ \)Pencil, you could use"sqrt" to indicate a square root like this in your expressions
if you don't have the square root symbol:

sqrt(5)[sqrt(15) + sqrt(5)]

Multiple Choice Answer:

A) 5^3 + ^5 \(\displaystyle \ \ \ \ \ \ \ \)5sqrt(3) + sqrt(5)

B) 25 ^3 \(\displaystyle \ \ \ \ \ \ \ \ \ \ \ \)25sqrt(3)

C) 3^5 + 5 \(\displaystyle \ \ \ \ \ \ \ \)3sqrt(5) + 5

D) 5^3 + 5 \(\displaystyle \ \ \ \ \ \ \ \)5sqrt(3) + 5

My (Pencil's) Approach:

^5(^15+^5) \(\displaystyle \ \ \ \ \ \ \ \)sqrt(5)(sqrt(15) + sqrt(5))

= ^5 ^20 \(\displaystyle \ \ \ \ \ \ \ \ \ \ \)sqrt(5)sqrt(20) \(\displaystyle \ \ \ \ \)\(\displaystyle \ \ \sqrt{15} + \sqrt{5} \ne \sqrt{20}\)

= ^5 ^4 ^5 \(\displaystyle \ \ \ \ \ \ \ \)sqrt(5)sqrt(4)sqrt(5)\(\displaystyle \ \ \ \ \ \)So the remaining steps do not follow.

= ^5 2 ^5 \(\displaystyle \ \ \ \ \ \ \ \ \ \)sqrt(5)2sqrt(5)

Clearly my approach was not part of the multiple choice answer so any help would be appreciated. Thanks again.

Pencil, look at this:

sqrt(5)[sqrt(15) + sqrt(5)] =

sqrt(5)*sqrt(15) + sqrt(5)*sqrt(5) =

sqrt(5)*sqrt(5)*sqrt(3) + 5 =


You can finish it.

 

[Pencil]

I figured out what it means.



Pencil, look at this:

sqrt(5)[sqrt(15) + sqrt(5)] =

sqrt(5)*sqrt(15) + sqrt(5)*sqrt(5) =

sqrt(5)*sqrt(5)*sqrt(3) + 5 =


You can finish it.

Yes, thank you. I was trying to use the square root symbol but was unable to copy and paste it without it looking weird... Anyway my apologies for using the exponent key as the radical sign. It wasn't my intention... With your assistance, I was able to simplify it to 25 sqrt(3), however I don't understand how you got sqrt(3)... on the last line. "sqrt(5)*sqrt(5)*sqrt(3) + 5 =" .... it seems to me, like you're using distributive property, but I don't see what happens to sqrt(15) on the 2nd line. "sqrt(5)*sqrt(15) + sqrt(5)*sqrt(5) =" but I proceeded from where you left off anyway, and got 25 sqrt (3).

I was trying to upload a jpeg of my approach at simplifying it, but it wouldn't let me... so here it goes:

sqrt(25) * sqrt(3) + 5 =
sqrt(75) + 5 =
5 sqrt(75) =
5 sqrt(25) sqrt(3) =
5 sqrt(5) sqrt(3) =
25 sqrt(3)

 

[Deleted member 4993]

Yes, thank you. I was trying to use the square root symbol but was unable to copy and paste it without it looking weird... Anyway my apologies for using the exponent key as the radical sign. It wasn't my intention... With your assistance, I was able to simplify it to 25 sqrt(3), however I don't understand how you got sqrt(3)... on the last line. "sqrt(5)*sqrt(5)*sqrt(3) + 5 =" .... it seems to me, like you're using distributive property, but I don't see what happens to sqrt(15) on the 2nd line. "sqrt(5)*sqrt(15) + sqrt(5)*sqrt(5) =" but I proceeded from where you left off anyway, and got 25 sqrt (3).

I was trying to upload a jpeg of my approach at simplifying it, but it wouldn't let me... so here it goes:

sqrt(25) * sqrt(3) + 5 =
sqrt(75) + 5 =
5 sqrt(75) =...........................Incorrect
5 sqrt(25) sqrt(3) =
5 sqrt(5) sqrt(3) =
25 sqrt(3)

sqrt(25) * sqrt(3) + 5 =
sqrt(75) + 5 =

sqrt(25*3) + 5 =

sqrt(25)*sqrt(3) + 5 =

5*sqrt(3) + 5 =

5 * [ 1 + sqrt(3)] ....................Answer

 

[Pencil]

RULE: SQRT(a * b) = SQRT(a) * SQRT(b) : tattoo this on your wrist

SQRT(15) = SQRT(5 * 3) = SQRT(5) * SQRT(3)

Wow thanks for that rule!
Really helpful.

 

[Pencil]

sqrt(25) * sqrt(3) + 5 =
sqrt(75) + 5 =

sqrt(25*3) + 5 =

sqrt(25)*sqrt(3) + 5 =

5*sqrt(3) + 5 =

5 * [ 1 + sqrt(3)] ....................Answer

Thanks! I definitely need more exposure to those types of problems but now I see concept.