what is square root

Question

how do I solve √9y = 49

 

 

 

how do I solve (4-3√2)^2, please show steps

Answer 1

You can think of a squareroot as the opposite of squaring:

 

2² = 4

so,

√4 = 2

 

other examples:

3² = 9,  √9 = 3        4² = 16, √16 = 4        5² = 25,  √25 = 5

 

        side note: unlike numbers being squared, the number under the squareroot can never be negative.          You can't have √(-9) for example.

 

using that logic here are some helpful tricks:

√(x²) = x and (√x)² = x

        examples: 

     √(3²) = 3  and (√3)² = 3

     √(10²) = 10 and (√10)² = 10

     √(1²) = 1 and (√1)² = 1

 

with the problem √9y = 49, how we solve this problem is dependent on whether or not the y is under the radical: is it √(9y) = 49  or (√9)y = 49?

if it is (√9)y = 49:

first we simplify the radical:

    √9 = √(3*3) = √(3²) = 3

thus,

   (√9)y = 49  =>  3y = 49  => y = 49/3.

if it is √(9y) = 49:

we square to get rid of the radical: (remember what ever you do to one side of the equation you have to do to the other side!!)

√(9y) = 49  =>  (√(9y))² = (49)²  =>  9y = (49)²  =>  9y = 2401  =>  y = 2401/9.

 

 

As for the second question:

 (4-3√2)^2

we FOIL: FOIL stands for First, Outer, Inner, Last:

I will show the steps as we go through them.

  (4-3√2)^2 = (4-3√2) x (4-3√2).

step 1: FIRST

(4-3√2) x (4-3√2)

 ^             ^

 4 x 4 = 16   

step 2: OUTER

(4-3√2) x (4-3√2)

 ^                  ^

4 x -3√2 = 4 x -3 x √2 = -12 x √2 = -12√2

step 3: INNER

(4-3√2) x (4-3√2)

      ^        ^

-3√2 x 4 = -3 x √2 x 4 = -12 x √2 = -12√2

step 4: LAST

(4-3√2) x (4-3√2)

      ^             ^

-3√2 x -3√2 = -3 x √2 x -3 x √2 = (-3)(-3) x √2 x √2 = 9 x √2 x √2 = 9 x (√2)(√2) = 9 x (√2)² = 9 x 2 = 18.  note: whenever multiplying squareroots together you can combine them under the same squareroot: i.e. √4 x √5 = √(4 x 5).

 

the last step is to  add the together the results:

(4-3√2) x (4-3√2) = 16 + -12√2 + -12√2 + 18 = 34 - 12√2 - 12√2.

because -12√2 and -12√2 are multiplied by the same squareroot number √2, we can use the distributive property:

34   - 12√2 - 12√2  = 34 +   √2 x (-12 -12) = 34 +   √2 x (144) =   34 +  144√2. 

we can't simplify any more. so that is the answer. 

 

Answer 2

A square root is the set or list of number that when multiplied by themselves results in the number inside the square root expression. When a radical symbol is used, it implies what is called the "principal square root", in other words the answer most like a positive number. So square roots of 4 are 2 and (-2); square roots of 9 are (3) and (-3). However, the principal square root of 4, written √4, is 2. The principal square root of 9, written √9, is 3. If you are looking for the general square root of 4, you would write it as a power: (4)^(1/2) or 4^.5, taking note of the decimal point because .5 = 1/2.

 

The first problem has already been answered quite clearly.

 

I feel that I can give a very clear piece of advice for your second problem:

 

Remember that x² = (x)(x). Therefore, (4-3√2)² =(4-3√2)(4-3√2). Foiling this should not be a problem, but to be clear, we have:

 

(4-3√2)(4-3√2)=(4)(4)+(4)(-3√2)+(-3√2)(4)+(-3√2)(-3√2)

 

=16 - 12√2 - 12√2 + 18

 

I'm sure you can finish it and fill in the missing steps.

Answer 3

It depends on whether it is the square root of 9 or the square root of 9y (I can't tell the way you have written it).

If it is the square root of 9 times y=49, then the solution is 3y= 49 and y=16.3

If it is the square root of 9y, then the solution is 9y equals 49 squared or 2401, and y=266.78

 

Can't help you with the other one. I don't know what ^ stands for.