SOLUTION: Can you please help me with these questions? a) sq. rt. of 12 plus sq.rt. of 108 b) ^3 sq. rt. of 16 times ^3 sq. rt. of 3 Thank you so much for your help.

Question 86391: Can you please help me with these questions?
a) sq. rt. of 12 plus sq.rt. of 108
b) ^3 sq. rt. of 16 times ^3 sq. rt. of 3
Thank you so much for your help.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)

First lets simplify :

Start with the given expression
The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.
So let's list the factors of 12
Factors:
1, 2, 3, 4, 6,

Notice how 4 is the largest perfect square, so lets break 12 down into 4*3

Factor 12 into 4*3

Break up the square roots using the identity

Take the square root of the perfect square 4 to get 2

So the expression

simplifies to

---------------------------------------------------------------------------
Now lets simplify :

Start with the given expression
The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.
So let's list the factors of 108
Factors:
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54,

Notice how 36 is the largest perfect square, so lets break 108 down into 36*3

Factor 108 into 36*3

Break up the square roots using the identity

Take the square root of the perfect square 36 to get 6

So the expression

simplifies to

So the expression

simplifies to

Notice we have a common term of . If we let we get

Now combine like terms

}

Replace y with

Check:

Evaluate the given expression with a calculator:

Evaluate the simplified expression with a calculator:

Since they are equal (to a certain decimal place), this verifies our answer

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b)

Since the 2 radicands are under the same root value, we can combine them using the identity

Combine the cube roots

Multiply

Factor 48 into 8*6. I chose to factor out an 8 since 8 is a perfect cube

Break up the roots using

Take the cube root of 8

Check:

Evaluate the given expression with a calculator:

Evaluate the simplified expression with a calculator:

Since they are equal (to a certain decimal place), this verifies our answer
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