Assign 8

course Mth 158

10-02-09 11:17 a.m.This chapter was extremely hard for me to understand.

If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution:

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

008. `* 8

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Question: * R.8.12. Simplify the cube root of 54

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Your solution:

The cube root of 54

54^(1/3) = cube root of 2 * 27 = cube root of 2 * 3^3

(I’m not sure what rule this is, but I know you can remove the 3’s since they are in groups of 3 and bring it to the outside saying)

3 * cube root of 2

confidence rating: 3

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Given Solution:

* * The cube root of 54 is expressed as 54^(1/3).

The number 54 factors into 2 * 3 * 3 * 3, i.e., 2 * 3^3. Thus

54^(1/3) = (2 * 3^3) ^(1/3)

= 2^(1/3) * (3^3)^(1/3)

= 2^(1/3) * 3^(3 * 1/3)

= 2^(1/3) * 3^1

= 3 * 2^(1/3), i.e.,

3 * cube root of 2.

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Self-critique (if necessary):

If I study the typewritten notation and actually put it on paper, I can see how to do it. When I try to type it out myself, it becomes extremely difficult for me to follow. I just typed (the cube root of..) instead of 2^(1/3). The answer is the same so there are no discrepancies between the two that I can see other than the way I typed mine.

You should of course continue working on fractional exponents, which provide a more direct application of the laws of exponents than do radicals.

However your notation was fine, and was completely equivalent to the given solution.

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Self-critique Rating: 3

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Question: * R.8.18. Simplify the cube root of (3 x y^2 / (81 x^4 y^2) ).

cube root of (3xy^2 / (81x^4y^2))

(This is one that the tutor showed me how to do. I can’t explain it the way you would do it, but here is my version of solving the problem the way I understand it)

Take the variables x y and solve for them first

x*y*y / x*x*x*x*y*y and seeing here that you can cancel x/x and yy/yy leaving the x^3

cube root of 3/81x^3 reducing the 81 to 3^4 then cancelling the 3/3 leaving

cube root of 1/3^3x^3 reducing further to 1/(3x)

confidence rating: 3

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Given Solution:

The cube root of (3 x y^2 / (81 x^4 y^2) ) is

(3 x y^2 / (81 x^4 y^2) ) ^ (1/3) =

(1 / (27 x^3) ) ^(1/3) =

1 / ( (27)^(1/3) * ^x^3^(1/3) ) =

1 / ( (3^3)^(1/3) * (x^3)^(1/3) ) =

1 / ( 3^(3 * 1/3) * x^(3 * 1/3) ) =

1 / (3 * x) =

1 / (3x).

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Self-critique (if necessary):

I know that my way is the long way around the given solution, but for the moment, it’s the only way I can understand. Since the answer is the same, there are no discrepancies between the two.

Again your explanation was fine, but keep working on fractional exponents.

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Self-critique Rating: 3

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Question: * R.8.30. Simplify 2 sqrt(12) - 3 sqrt(27).

2 sqrt(12) – 3 sqrt(27) reducing the first

2 sqrt(3) – 3 sqrt(27) then reducing the second to

2 sqrt(3) – 3 sqrt(3) here I just multiplied 2 * sqrt to get 4 sqrt(3) and then 3 * cube root to get 9 sqrt (3)

(this step is hard to explain on paper, but this is the way the tutor showed me and it comes out to the correct answer)

4 sqrt(3) – 9 sqrt(3) then subtracting 4-9

-5 sqrt(3)

confidence rating: 3

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Given Solution:

2 sqrt(12) - 3 sqrt(27)

= 2 sqrt( 2*2*3) - 3 sqrt(3*3*3)

= 2 sqrt(2^2 * 3) - 3 sqrt(3^3)

= 2 sqrt(2^2) sqrt(3) - 3 sqrt(3^2) sqrt(3)

= (2 * 2 - 3 * 3) sqrt(3)

= (4 - 9) sqrt(3)

= -5 sqrt(3)

I don’t see how you get 2 * 2 – 3 * 3 sqrt(3) But I can see from this step how you obtain the last step.

The given solution didn't have the signs of grouping right; I've corrected them on the original page and in the solution as given here.

Extra Question: What is the simplified form of (2 sqrt(6) + 3) ( 3 sqrt(6)) and how did you get this result?

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Your solution:

(2 sqrt(6) + 3) ( 3 sqrt(6)) using the foil method, I multiplied 2 * 3 and sqrt6 * sqrt6

6(sqrt(6))^2 + (3*3 sqrt(6))

(6*6) + 9 sqrt(6)

36 + 9 sqrt(6)

confidence rating: 3

FOIL doesn't apply at all here. You don't have two binomials to which to apply it.

You used the distributive law (correctly).

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Given Solution:

(2*sqrt(6) +3)(3*sqrt(6)) expands by the Distributive Law to give

(2*sqrt(6) * 3sqrt(6) + 3*3sqrt(6)), which we rewrite as

(2*3)(sqrt6*sqrt6) + 9 sqrt(6) =

(6*6) + 9sqrt(6) =

36 +9sqrt(6).

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Self-critique (if necessary):

I don’t think there are any discrepancies other than Distributive law and FOIL method.

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Self-critique Rating: 3

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Question: * R.8. Expand (sqrt(x) + sqrt(5) )^2

(sqrt(x) + sqrt(5) )^2

= (sqrt(x) + sqrt(5) ) * (sqrt(x) + sqrt(5) )

= sqrt(x) * (sqrt(x) + sqrt(5) ) + sqrt(5) * (sqrt(x) + sqrt(5) ) then using the distributive law

= sqrt(x) * sqrt(x) + sqrt(x) * sqrt(5) + sqrt(5) * sqrt(x) + sqrt(5) * sqrt(5)

Then I get lost

confidence rating: 0

sqrt(5) * sqrt(5) = 5 and

sqrt(x) * sqrt(x) = x.

That's what the square root of a number means--the number which when multiplied by itself gives you the original number.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

(sqrt(x) + sqrt(5) )^2

= (sqrt(x) + sqrt(5) ) * (sqrt(x) + sqrt(5) )

= sqrt(x) * (sqrt(x) + sqrt(5) ) + sqrt(5) * (sqrt(x) + sqrt(5) )

= sqrt(x) * sqrt(x) + sqrt(x) * sqrt(5) + sqrt(5) * sqrt(x) + sqrt(5) * sqrt(5)

= x + 2 sqrt(x) sqrt(5) + 5.

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Self-critique (if necessary):

I can follow along until the last line, and I can’t see how to get x + 2sqrt(x) sqrt(5) + 5

It just doesn’t make any sense to me.

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Self-critique Rating: 0

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Question:

* R.8.42. What do you get when you rationalize the denominator of 3 / sqrt(2) and what steps did you follow to get this result?

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Your solution:

3/ sqrt(2) first you multiply the entire problem by the denominator

3/ sqrt(2) * sqrt(2)/sqrt(2)

3sqrt(2) / sqrt(2)^2

3 sqrt(2) / 2

confidence rating: 3

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Given Solution:

Starting with 3/sqrt(2) we multiply numerator and denominator by sqrt(2) to get

(3*sqrt(2))/(sqrt(2)*sqrt(2)) =

(3 sqrt(2) ) /2.

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Self-critique (if necessary):

The only thing I see is the given solution starts out with (2 * sqrt(2)) where I have 3 sqrt(2).

There was a typo in the given solution, which I've corrected.

Your solution was good.

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Self-critique Rating: 3

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Question: * R.8.48. Rationalize denominator of sqrt(2) / (sqrt(7) + 2)

First you multiply by the denominator, by changing the + to a – when you flip it

( sqrt(2) / (sqrt(7) + 2) ) * (sqrt(7) - 2) / (sqrt(7) - 2)

= sqrt(2) * (sqrt(7) - 2) / ( (sqrt(7) + 2) * ( sqrt(7) - 2) )

= sqrt(2) * (sqrt(7) - 2) / (sqrt(7) * sqrt(7) - 4)

= sqrt(2) * (sqrt(7) - 2 ) / (7 - 4)

= sqrt(2) * (sqrt(7) - 2 ) / 3.

To rationalize the denominator sqrt(7) + 2 we multiply both numerator and denominator by sqrt(7) - 2.

We obtain

( sqrt(2) / (sqrt(7) + 2) ) * (sqrt(7) - 2) / (sqrt(7) - 2)

= sqrt(2) * (sqrt(7) - 2) / ( (sqrt(7) + 2) * ( sqrt(7) - 2) )

= sqrt(2) * (sqrt(7) - 2) / (sqrt(7) * sqrt(7) - 4)

= sqrt(2) * (sqrt(7) - 2 ) / (7 - 4)

= sqrt(2) * (sqrt(7) - 2 ) / 3.

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Self-critique (if necessary):

This one was a little more difficult to follow along.

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Self-critique Rating: 3

Extra Question: What steps did you follow to simplify (x^3)^(1/6) and what is your result, assuming that x is positive and expressing your result with only positive exponents?

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Your solution:

(x^3)^(1/6) I took the power of 3 and turned it into a fraction and multiplied it by the 1/6 to get

x^(1/3 * 1/6) to simplify to x^(1/2)

confidence rating: 3

right idea but x^3 is not the same thing as x^(1/3).

note that 1/3 * 1/6 would be 1/18

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Given Solution:

* * Express radicals as exponents and use the laws of exponents.

(x^3)^(1/6) =

x^(3 * 1/6) =

x^(1/2). **

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Self-critique (if necessary):

There are no discrepancies.

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Self-critique Rating: 3

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Question: * R.8.60. Simplify 25^(3/2).

25^(3/2) Here the 3 would represent the power and the 2 is the sqrt of 25. To solve, you would multiply 3/1 * ½

I don’t think that is right because I just come up with the same thing.

confidence rating: 0

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Given Solution:

25^(3/2) =

(5^2)^(3/2) =

5^(2 * 3/2) =

5^(2 * 3/2) =

5^3.

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Self-critique (if necessary):

I see now how you solved. You took the sqrt of 25 raised to the power of 2 and multiplied the 2/1 * 3/2 to get 5^3

Good.

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Self-critique Rating: 3

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Question: * R.8.72. Simplify and express with only positive exponents:

(xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4).

I really don’t know how to solve this one on my own. But I will try.

(xy)^(1/4) (x^2 y^2)^(1/2) / (x^2y)^(3/4)

First we separate the x and y

(x)^(1/4) * (y)^(1/4) * sqrt(x^2) * sqrt(y^2) / (x^2)^(3/4) * y^(3/4)

(x)^(1/4) * (y)^(1/4) * (x^(2 * ½) * y^(2 * ½) / x^(2 * ¾) * y^(3/4)

(x)^(1/4) * (y)^(1/4) * xy / (x)^(3/2) * (y)^(3/4)

Then I don’t know what to do from here.

confidence rating: 0

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Given Solution:

(xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4)

= x^(1/4) * y^(1/4) * (x^2)^(1/2) * y^2 ^ (1/2) / ( (x^2)^(3/4) * y^(3/4) )

= x^(1/4) * y^(1/4) * x^(2 * 1/2) * y^(2 * 1/2) / ( (x^(2 * 3/4) * y^(3/4) )

= x^(1/4) y^(1/4) * x^1 * y^1 / (x^(3/2) y^(3/4) )

= x^(1 + 1/4) y^(1 + 1/4) / (x^(3/2) y^(3/4) )

= x^(5/4) y^(5/4) / (x^(3/2) y^(3/4) )

= x^(5/4 - 3/2) y^(5/4 - 3/4)

= x^(5/4 - 6/4) y^(2/4)

= x^(-1/4) y^(1/2)

= y^(1/2) / x^(1/4).

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Self-critique (if necessary):

I wrote the entire given solution on paper to see how to solve, but I am still confused when it gets to the = x^(1 + 1/4) y^(1 + 1/4) / (x^(3/2) y^(3/4)

How do you get 1 + ¼? Does the 1 come from the xy on the right of the numerator?

The numerator of the expression

x^(1/4) y^(1/4) * x^1 * y^1 / (x^(3/2) y^(3/4) )

contains two factors which are powers of x. The two are

x^(1/4) and x^1 (the latter could be written just as x, but to apply the laws of exponents it's not a bad idea to write the exponent explicitly).

When you multiply these two factors, the laws of exponent tell you that you get x^(1/4 + 1) = x^(5/4).

The same thing happens with the y factors.

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Self-critique Rating: 2

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Question: * R.8.84. Express with positive exponents:

( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2).

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Your Solution:

confidence rating: 0

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2)

=

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Self-critique (if necessary):

I still don’t know what this is about? Is there anything different between the given solution and the question?

That solution got cut off.

Here it is:

( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2) =

( (9 - x^2) ^(1/2) + x^2 / ( 9 - x^2) ^(1/2) ) / (9 - x^2) =

(9 - x^2) (1/2) / (9 - x^2)^1 + x^2 / ( (9 - x^2)^(1/2) * (9 - x^2)^ 1) =

(9 - x^2) ^(-1/2) + x^2 / (9 - x^2)^(3/2) =

1 / (9 - x^2)^(1/2) + x^2 / (9 - x^2)^(3/2).

In the third step the exponent ^1 on the (9 - x^2) expressions wasn't necessary, but was included to explicitly show the exponents and the application of the laws of exponents.

The first term in the 4th step is obtained as follows:

(9 - x^2) (1/2) / (9 - x^2)^1 = (9 - x^2) ^ (1/2 - 1) = (9 - x^2)^(-1/2).

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Self-critique Rating: 0

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Question: * R.8.108. v = sqrt(64 h + v0^2); find v for init vel 0 height 4 ft; for init vel 0 and ht 16 ft; for init vel 4 ft / s and height 2 ft.

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Your Solution:

v = sqrt(64 h + v0^2)

If initial velocity = 0 and height = 4 ft then we substitute v0 = 0 and h = 4

v = sqrt(64 * 4 + 0^2) = sqrt(256) =16.

If initial velocity is 0 and height is 16 ft then we substitute v0 = 0 and h = 16

v = sqrt(64 * 16 + 0^2) = sqrt(1024) = 32.

If initial velocity is 4 ft/s and height 2 ft.

v = sqrt(64 * 2 + 4^2) = sqrt(144) = 12 (this can’t be right, but I don’t know how to do it…what is 4ft/s?

confidence rating: 2

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Given Solution:

If initial velocity is 0 and height is 4 ft then we substitute v0 = 0 and h = 4 to obtain

• v = sqrt(64 * 4 + 0^2) = sqrt(256) =16.

If initial velocity is 0 and height is 16 ft then we substitute v0 = 0 and h = 4 to obtain

• v = sqrt(64 * 16 + 0^2) = sqrt(1024) = 32.

Note that 4 times the height results in only double the velocity.

If initial velocity is 4 ft / s and height is 2 ft then we substitute v0 = 4 and h = 2 to obtain

• v = sqrt(64 * 2 + 4^2) = sqrt(144) =12.

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Self-critique (if necessary):

I got the answer right, but I don’t know how I did it other than just substituting the given numbers for velocity and height.

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Self-critique Rating: 1

Extra Question: What is the simplified form of (24)^(1/3) and how did you get this result?

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Your solution:

cube root of 24 simplifies as

cube root of (8 * 3)

then you can take the

cube root of 8 and multiply by the cube root of 3

When you do this, the cube root of 8 becomes

8 = 2 * 4 = 2 * 2 * 2, which leaves us with 2 since there are 3 2’s

2 * cube root of 3 or 2 * 3

confidence rating: 2

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

* * (24)^(1/3) =

(8 * 3)^(1/3) =

8^(1/3) * 3^(1/3) =

2 * 3^(1/3) **

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Self-critique (if necessary):

I am still at the point of putting it on paper and actually doing 2*2*2 to see it. I guess visuals help! Does the cube root of 3 just mean 3? and would my answer of 2 * 3 be correct?

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Self-critique Rating: 2

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Question:

Extra Question: What is the simplified form of (x^2 y)^(1/3) * (125 x^3)^(1/3) / ( 8 x^3 y^4)^(1/3) and how did you get this result?

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Your solution:

I worked this on paper and was sure I done it right, but it wasn’t. Here is the way I did it.

cube root x^2y * cute root 125x^3 / cube root 8x^3y^4

then I used the distributive law to get

cube root 5^3x^3y * cube root 8x^3y^4 / cube root 8x^3y^4

you wouldn't multiply and then divide by cube root 8x^3y^4 ; you started with only cube root 8x^3y^4 in the numerator.

then eliminated the 5^3 on the left and the x^3 on both sides and the y^4 on the top and bottom to leave

cube root 2^3 / cube root 2^3x^3

It is all wrong and I just get so confused on how to do this.

confidence rating: 0

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Given Solution:

* * (x^2y)^(1/3) * (125x^3)^(1/3)/ ( 8 x^3y^4)^(1/3)

(x^(2/3)y^(1/3)* (5x)/[ 8^(1/3) * xy(y^(1/3)]

(x^(2/3)(5x) / ( 2 xy)

5( x^(5/3)) / ( 2 xy)

5x(x^(2/3)) / ( 2 xy)

5 ( x^(2/3) ) / (2 y) **

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

I can’t remember what law goes with what problem and I am trying to solve all of the the same way. I just don’t understand this at all.

you understand it pretty well, but not completely

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Self-critique Rating: 0

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Question: Extra question. What is the simplified form of sqrt( 4 ( x+4)^2 ) and how did you get this result?

Another one I just can’t solve.

confidence rating:

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Given Solution:

We use two ideas in this solution:

• sqrt(a b) = sqrt(a) * sqrt(b) and

• sqrt(x^2) = | x |

To understand why sqrt(x^2) = | x | and not just x consider the following:

• Let x = 5. Then sqrt(x^2) = sqrt( 5^2 ) = sqrt(25) = 5, so sqrt(x^2) = x.

It is also clear that in this case, | x | = | 5 | = 5, so | x | = x, and we can say that sqrt(x^2) = | x |.

• Now let x = -5. We get sqrt(x^2) = sqrt( (-5)^2 ) = sqrt(25) = 5.

In this case sqrt(x^2) = 5 but x is not equal to 5, so sqrt(x^2) is not x.

However, in this case | x | = | -5 | = 5, so it is the case the sqrt(x^2) = | x |.

Using these ideas we get

• sqrt( 4 ( x+4)^2 ) = sqrt(4) * sqrt( (x+4)^2 ) = 2 * | x+4 | **

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Self-critique (if necessary):

The given solution confuses me even more on this one.

&#You did not answer the given question. You need to always at least explain what you do and do not understand about the question. A phrase-by-phrase analysis is generally required when you cannot otherwise answer a question.

&#

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Self-critique Rating:

* Add comments on any surprises or insights you experienced as a result of this assignment.

You're doing OK here. You haven't quite put it all together, so you might miss something on the test, but I think you'll get more problems related to this section right than wrong.