Explanation

The set {x: x ∈ N, x ≤ 10} has 10 elements (1, 2, 3, 4, 5, 6, 7, 8, 9, 10).

The cardinality of the power set of a set with n elements is 2^n.

So, the cardinality of the power set is 2^10 = 1024.

Explanation

${x^}^2+18x+81=0$:

• The first term is x^2, so $a=x$.
• The last term is 81, so ${b^}^2=81$, which means $b=9$ (we take the positive root since the middle term is positive).
• The middle term is 18x. Let's check if it matches 2ab: $2(x)(9)=18x$. It does.

Therefore, the quadratic expression ${x^}^2+18x+81$ can be factored into $(x+9{)^}^2$.

So the factor form of the equation ${x^}^2+18x+81=0$ is $(x+9{)^}^2=0$.

Explanation

step 2:

Explanation

12/35

sum of numbers = 12

Product of numbers =35

Let the numbers be x and y

x+ y= 12

xy= 35

According to question we have to find 1/x + 1/y

 1/x + 1/y =(y+ x)/xy

 1/x + 1/y=12/35

sum of their recriprocal is 12/35

Explanation

The given partial differential equation:

∂²U/∂x² + ∂²U/∂y² = f(x,y)

is known as the Poisson equation.

If f(x,y) = 0, it would reduce to the Laplace equation.

Explanation

To integrate √(2x + 1), we can use the power rule of integration and the chain rule.

Let u = 2x + 1, then du/dx = 2, and du = 2dx.

∫√(2x + 1) dx = (1/2) ∫u^(1/2) du

= (1/2) (2/3) u^(3/2) + C

= (1/3) (2x + 1)^(3/2) + C

Explanation

To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Calculate the square roots:

√10 ≈ 3.162

√250 = √(25 × 10) = 5√10 ≈ 5 × 3.162 = 15.811

2. Multiply the square roots:

3.162 × 15.811 ≈ 50.00

Explanation

Explanation

Step 1: Evaluate the integral

∫(x² - x - 1) dx = (x³ / 3) - (x² / 2) - x

Step 2: Apply the limits of integration

[(1³ / 3) - (1² / 2) - 1] - [((-1)³ / 3) - ((-1)² / 2) - (-1)]

= [(1/3) - (1/2) - 1] - [(-1/3) - (1/2) + 1]

Step 3: Simplify the expression

= (1/3) - (1/2) - 1 + (1/3) + (1/2) - 1

= (2/3) - 2

= (2/3) - (6/3)

= -4/3

Explanation

Given:

3x : 2y = 6 : 5

Cross-multiply:

(3x) × 5 = (2y) × 6

15x = 12y

Divide both sides by 3y:

5x/y = 4

x/y = 4/5

So, x : y = 4 : 5