Simplifying each of the following expressions:
a) (√5-√12)²:
First, we can simplify the square of each term using the formula (a-b)² = a² - 2ab + b²:
(√5-√12)² = (√5)² - 2(√5)(√12) + (√12)²
= 5 - 2√60 + 12
Next, we can simplify √60 by factoring out its largest perfect square factor, which is 4:
5 - 2√60 + 12 = 17 - 2√(4*15) = 17 - 4√15
Therefore, (√5-√12)² = 17 - 4√15.
b) (√12+√11)²:
We can simplify this expression using the same formula as before:
(√12+√11)² = (√12)² + 2(√12)(√11) + (√11)²
= 12 + 2√132 + 11
Next, we can simplify √132 by factoring out its largest perfect square factor, which is 4:
12 + 2√132 + 11 = 23 + 2√(4*33) = 23 + 2√132
Therefore, (√12+√11)² = 23 + 2√132.
c) (√6+√7) (√6-√7):
This expression can be simplified using the difference of squares formula, (a-b)(a+b) = a² - b²:
(√6+√7) (√6-√7) = (√6)² - (√7)²
= 6 - 7
= -1
Therefore, (√6+√7) (√6-√7) = -1.
Rationalizing each of the following expressions:
a) 1/√8:
To rationalize the denominator, we can multiply both the numerator and denominator by √8:
1/√8 = (1/√8) * (√8/√8) = √8/8
Therefore, 1/√8 = √8/8.
b) 1/√7-√1:
To rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator, which is √7+√1:
1/√7-√1 = (1/√7-√1) * (√7+√1)/(√7+√1) = (√7+√1)/(7-1)
= (√7+√1)/6
Therefore, 1/√7-√1 = (√7+√1)/6.
c) 1/√8+3:
To rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator, which is √8-3:
1/√8+3 = (1/√8+3) * (√8-3)/(√8-3) = (√8-3)/(8-9)
= -(√8-3)
Next, we can simplify √8 by factoring out its largest perfect square factor, which is 4:
-(√8-3) = -