Respuesta :
Answer:
3a. horizontal shift to the left 4 units, reflection on the x-axis
3b. vertical shift down 5 units
4. 4[tex]\sqrt{3}[/tex]
5. 3x²y²[tex]\sqrt{2x}[/tex]
6. [tex]\frac{4\sqrt{35} }{5}[/tex]
7. [tex]\frac{45-5\sqrt{7} }{74}[/tex]
8. 3√2
9. -2√7 + 5√13
10. 7√2 + 2√14
11. 10 + √15 + 2√35 + √21
12. P = 18√6 + 6√15
Step-by-step explanation:
this is gonna be a bit long but bare with me
3a. y = [tex]-\sqrt{x+4}[/tex] < -- this is a horizontal shift to the left 4 units and a reflection on the x-axis. a horizontal shift is located next to the parent function (in this case f(x) = √x). a reflection on the x-axis is located outside of the parent function.
3b. y = [tex]\sqrt{x}[/tex] - 5 < --- this is a vertical shift down 5 units. a vertical shift is located outside of the parent function
4. [tex]\sqrt{48}[/tex] to solve a radicand, we can break it up into known factors, such as numbers that are a perfect square that are factors of the number. you can create a factor tree for this, or you can do it mentally by thinking of terms.
48 can be divided by divided by 16 and 3
16 is a perfect square of 4, which goes outside of the radicand
the answer to [tex]\sqrt{48}[/tex] is 4[tex]\sqrt{3}[/tex]
5. again, to solve [tex]\sqrt{18x^5y^4}[/tex], we break it down into known factors.
lets start with 18: it can be broken down into 9 and 2, 9 is a perfect square of 3
so far the radicand looks like this: 3[tex]\sqrt{2x^5y^4}[/tex]
next we will break down [tex]x^5[/tex]. looking at the exponent, we can see that we have a perfect square in the exponent, which is 4. the square root of 4 is 2. we would leave the leftover x in the radical as it is not a perfect square, and put x² on the outside as it is a result of the perfect square
3x²[tex]\sqrt{2xy^4}[/tex]
finally lets look at [tex]y^4[/tex]. we see that the exponent is a perfect square, 4. the square root of 4 is 2, so y² would go on the outside, and no y terms would be left on the inside. the final radical looks like this:
3x²y²[tex]\sqrt{2x}[/tex]
6. [tex]\frac{4\sqrt{7} }{\sqrt{5} }[/tex]
to solve this, we would need to rationalize the denominator by getting rid of the radical in the denominator. to do this we would multiply the expression by [tex]\sqrt{5}[/tex] and we would get:
[tex]\frac{4\sqrt{35} }{5}[/tex]
we cannot simplify this any further, so this would be our answer
7. [tex]\frac{5}{9 + \sqrt{7} }[/tex] like the last question, we need to get rid of the square root in the denominator by multiplying the entire expression by [tex]9 - \sqrt{7}[/tex]. we get:
[tex]\frac{5(9-\sqrt{7}) }{(9+\sqrt{7})(9-\sqrt{7}) }[/tex]
the denominator can be multiplied together to get 81 - 7 = 74, and in the numerator we get 5(9-√7) which can be distributed as 45 - 5√7. in total we get:
[tex]\frac{45-5\sqrt{7} }{74}[/tex] this cannot be simplified any further, so this is the answer
8. 3√50 - 3√32 to solve, we need to get the same radical. to do this, we can break each radical up into known terms. √50 can be simplified as 25 and 2. 25 is a perfect square, and is equal to 5. √50 = 5√2
3(5√2) = 15√2
we can break up √32 as 16 and 2. 16 is a perfect square which is equal to 4. √32 = 4√2
-3(4√2) = -12√2
the expression looks like this: 15√2 - 12√2 < subtract normally and we get 3√2 <-- this is our final answer, and we cannot simplify further
9. 2√7 - 4√13 -4√7 + 9√13 < we can combine like terms by adding/subtracting
2√7 -4√7 = -2√7
-4√13 + 9√13 = 5√13
-2√7 + 5√13 this cannot be simplified further, so this is the answer
10. √7(√14 + 2√2) < distribute √7 into the expression
√7 × √14 = √98 = 7√2
√7 × 2√2 = 2√14
the answer is 7√2 + 2√14
11. (√5 + √7)(√20 + √3) < to solve, we can FOIL this out
(√5 + √7)(√20 + √3) = 10
(√5 + √7)(√20 + √3) = √15
(√5 + √7)(√20 + √3) = 2√35
(√5 + √7)(√20 + √3) = √21
we get the following expression: 10 + √15 + 2√35 + √21, we cannot simplify this further so this is our answer
12. the perimeter of a rectangle is P = 2l + 2w. we are given the width 3√6 + 4√15 and the length 6√6 - √15. plugged into the formula we have:
2(3√6 + 4√15) + 2(6√6 - √15) < distribute 2 into each expression:
(6√6 + 8√15) + (12√6 - 2√15) < combine like terms, as in combine terms with like radicals
6√6 + 12√6 = 18√6
8√15 - 2√15 = 6√15
P = 18√6 + 6√15
we cannot simplify this further, so this is the answer
hope this helped!
