1)Rationalize the denominator;5-2√3÷3+3√3
2)Simplify the expression √3-√2÷√3+√2 giving your answer in the form of a+b√c
3)Simplify √96÷5+√24 giving your answer in the form of a+b√c
4)Simplify 2√5÷√3-√5 giving your answer in the form of a+b√c
5)Express 5/2√2-5+2/2√2+5 in the form of a+b√c
6)Express 1/√5+5/2√5-5 in the form of a+b√c
7)Find a,b and c if √28/2+√7-√28/√7=a+b√c
8)Evaluate (3-2√3)^2 hence rationalize the denominator in √3/(3-2√3)^2
9)Write 2√3/√2+√3 in the simplest form leaving your answer in the form of a+b√c
10)Simplify;√48/√5+√3 leaving your answer in the form of a+b√c
11)Simplify;4/5+√2-3/5-√2 giving your answer in surd form with a rational denominator
12)Express √3/1-cos30° in surd form and simplify
1) Rationalize the denominator:
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator.
(5 - 2√3)/(3 + 3√3) * (3 - 3√3)/(3 - 3√3)
This simplifies to:
(15 - 15√3 - 6√3 + 6√9)/(9 - 27)
(15 - 21√3)/(9 - 27)
Simplifying further:
(15 - 21√3)/(-18)
(-15 + 21√3)/18
Therefore, the rationalized form of the expression is (-15 + 21√3)/18.
2) Simplify the expression:
(√3 - √2)/(√3 + √2)
To simplify this expression, we need to rationalize the denominator.
(√3 - √2)/(√3 + √2) * (√3 - √2)/(√3 - √2)
This simplifies to:
(√3 - √2)(√3 - √2)/(3 - 2)
(3 - √6 - √6 + 2)/(1)
(5 - 2√6)/1
Therefore, the simplified form of the expression (√3 - √2)/(√3 + √2) is 5 - 2√6.
3) Simplify the expression:
√96/(5 + √24)
To simplify this expression, we rationalize the denominator.
√96/(5 + √24) * (5 - √24)/(5 - √24)
This simplifies to:
√96(5 - √24)/(25 - 24)
√96(5 - √24)/1
√96(5 - √24)
Therefore, the simplified form of the expression √96/(5 + √24) is √96(5 - √24).
4) Simplify the expression:
(2√5)/(√3 - √5)
To simplify this expression, we rationalize the denominator.
(2√5)/(√3 - √5) * (√3 + √5)/ (√3 + √5)
This simplifies to:
(2√15 + 2√5)/(3 - 5)
(2√15 + 2√5)/(-2)
-√15 - √5
Therefore, the simplified form of the expression (2√5)/(√3 - √5) is -√15 - √5.
5) Express the expression (5/2√2 - 5)/(2/2√2 + 5) in the form of a + b√c.
To simplify this expression, we first need to find a common denominator for both the numerator and denominator.
(5/2√2 - 5)/(2/2√2 + 5) * (√2/√2)
(5√2 - 5√2)/(2 - 5√2)
0/(-3√2)
0
Therefore, the expression (5/2√2 - 5)/(2/2√2 + 5) simplifies to 0.
6) Express the expression (1/√5 + 5/2√5 - 5) in the form of a + b√c.
To simplify this expression, we first need to find a common denominator for the fractions.
(1/√5 + 5/2√5 - 5) * (√5/√5)
((1√5 + 5√5)/√5 - 5√5)/√5
(6√5 - 5√5)/√5
√5/√5
1
Therefore, the expression (1/√5 + 5/2√5 - 5) simplifies to 1.
7) Find a, b, and c if (√28/2 + √7 - √28/√7) = a + b√c.
To solve for the values of a, b, and c, simplify the expression:
(√28/2 + √7 - √28/√7) * (√7/√7)
((√28√7 + √7)/2√7 - √28)/(√7)
Simplifying further:
((√196 + √7)/2√7 - √28)/(√7)
(14 + √7 - √196)/(2√7 - √28)/(√7)
(14 + √7 - 14)/(2 - 2)
√7/2
Therefore, a = 0, b = 1, and c = 7. The expression (√28/2 + √7 - √28/√7) can be written as 0 + √7√7.
8) Evaluate (3 - 2√3)^2 and rationalize the denominator in √3/(3 - 2√3)^2.
(3 - 2√3)^2
(3 - 2√3)(3 - 2√3)
9 - 6√3 - 6√3 + 12
21 - 12√3
To rationalize the denominator in √3/(3 - 2√3)^2, we multiply the numerator and denominator by the conjugate of the denominator:
√3/(3 - 2√3)^2 * (3 + 2√3)/(3 + 2√3)
(√3(3 + 2√3)/(9 - 12√3 + 12))
(3√3 + 2√9)/(9 - 12√3 + 12)
(3√3 + 6)/(21 - 12√3)
Therefore, the rationalized form of the expression is (3√3 + 6)/(21 - 12√3).
9) Write 2√3/(√2 + √3) in the simplest form, leaving the answer in the form of a + b√c.
To simplify this expression, we rationalize the denominator.
2√3/(√2 + √3) * (√2 - √3)/(√2 - √3)
(2√6 - 2√9)/(2 - √6)
(2√6 - 2√9)/(2 - √6) * (2 + √6)/(2 + √6)
(4√6 + 6 - 2√54 - √54)/(4 - 6)
(10√6 - 3√6 - 2√54)/(2)
(10√6 - 3√6 - √(2*27))/(2)
(10√6 - 3√6 - 3√2√3)/(2)
(10√6 - 3√6 - 3√6)/(2)
(4√6 - 6)/(2)
2√6 - 3
Therefore, the simplest form of the expression 2√3/(√2 + √3) is 2√6 - 3.
10) Simplify √48/(√5 + √3), leaving the answer in the form of a + b√c.
To simplify this expression, we first need to find a common denominator.
√48/(√5 + √3) * (√5 - √3)/(√5 - √3)
(√48√5 - √48√3)/(5 - 3)
(4√10 - 4√12)/(2)
2√10 - 2√12
Now, we can simplify further:
2√10 - 2√(4*3)
2√10 - 4√3
Therefore, the simplified form of the expression √48/(√5 + √3) is 2√10 - 4√3.
11) Simplify the expression (4/5 + √2 - 3/5 - √2), giving the answer in surd form with a rational denominator.
We can combine like terms together:
(4/5 - 3/5) + (√2 - √2)
(1/5) + (0)
1/5
Therefore, the expression (4/5 + √2 - 3/5 - √2) simplifies to 1/5.
12) Express √3/(1 - cos30°) in surd form and simplify.
To simplify this expression, we first need to find the value of cos30°.
cos30° = √3/2
Substituting the value into the expression:
√3/(1 - √3/2)
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
√3/(1 - √3/2) * (2/2)
(√3 * 2)/(2 - √3)
(2√3)/(2 - √3) * (2 + √3)/(2 + √3)
(2√9 + 3)/(4 - 3)
(2√9 + 3)/1
2√9 + 3
Therefore, the expression √3/(1 - cos30°) can be written as 2√9 + 3.