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3 years ago

Simplify the following expression, show all work you did to get the results

Mathematics

1 answer:

777dan777 [17]3 years ago

5 0

Answer:

(sqrt(7))/3 or decimal 0.8819171036881968635005385845464201419034197276941500601227781530...

Step-by-step explanation:

Simplify the following:

(sqrt(14))/(sqrt(18))

Hint: | Simplify radicals.

sqrt(18) = sqrt(2×3^2) = 3 sqrt(2):

(sqrt(14))/(3 sqrt(2))

Hint: | Multiply numerator and denominator of (sqrt(14))/(3 sqrt(2)) by sqrt(2).

Rationalize the denominator. (sqrt(14))/(3 sqrt(2)) = (sqrt(14))/(3 sqrt(2))×(sqrt(2))/(sqrt(2)) = (sqrt(14) sqrt(2))/(3×2):

(sqrt(14) sqrt(2))/(3×2)

Hint: | Multiply 3 and 2 together.

3×2 = 6:

(sqrt(14) sqrt(2))/6

Hint: | For a>=0, sqrt(a) sqrt(b) = sqrt(a b). Apply this to sqrt(14) sqrt(2).

sqrt(14) sqrt(2) = sqrt(14×2):

(sqrt(14×2))/6

Hint: | Multiply 14 and 2 together.

14×2 = 28:

(sqrt(28))/6

Hint: | Simplify radicals.

sqrt(28) = sqrt(2^2×7) = 2 sqrt(7):

(2 sqrt(7))/6

Hint: | In (2 sqrt(7))/6, divide 6 in the denominator by 2 in the numerator.

2/6 = 2/(2×3) = 1/3:

Answer: (sqrt(7))/3

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$f(x)=e^{x^2+2x}\implies f'(x)=2(x+1)e^{x^2+2x}$

$f$ has critical points where the derivative is 0:

$2(x+1)e^{x^2+2x}=0\implies x+1=0\implies x=-1$

The second derivative is

$f''(x)=2e^{x^2+2x}+4(x+1)^2e^{x^2+2x}=2(2x^2+4x+3)e^{x^2+2x}$

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