Explanation

A complex number is written in the form:

$$a+bi$$

• a is the real part

• b is the imaginary part

So in 2 − 4i:

• Real part = 2

• Imaginary part = −4

Explanation

 A complex number a + bi is considered purely imaginary if its real part, 'a', is equal to zero.

In other words, a pure imaginary number has the form bi, where 'b' is a real number. 

Explanation

Given points A(3, 1) and B(7, -5), let's calculate the distance using the distance formula:

Distance = √((x2 - x1)² + (y2 - y1)²)

= √((7 - 3)² + (-5 - 1)²)

= √((4)² + (-6)²)

= √(16 + 36)

= √52

= √(4 × 13)

= 2√13

Explanation

Given the area of the rectangle:

x² - 5x + 6

Let's factorize the quadratic expression:

x² - 5x + 6 = x² - 3x - 2x + 6

= x(x - 3) - 2(x - 3)

= (x - 3)(x - 2)

Explanation

Given:

(x, 6) = (2, y - 6)

Equating the corresponding components:

x = 2

6 = y - 6

Solving for y:

y = 6 + 6

y = 12

Now, finding x + y:

x + y = 2 + 12

= 14

Explanation

Given:

a < b

b < c

Using transitive property, if a < b and b < c, then a < c.

Explanation

Explanation

You gave the equation:  

xy² + xy = 8x  

We are to find dy/dx at the point (1, 2).

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Step 1: Differentiate both sides implicitly with respect to x:

d/dx(xy²) + d/dx(xy) = d/dx(8x)

Use product rule:  

d/dx(xy²) = x·d/dx(y²) + y²·d/dx(x) = x·2y·dy/dx + y²  

d/dx(xy) = x·dy/dx + y  

d/dx(8x) = 8

So the full derivative becomes:  

x·2y·dy/dx + y² + x·dy/dx + y = 8

Now plug in x = 1, y = 2:

1·2·2·dy/dx + 4 + 1·dy/dx + 2 = 8  

=> 4·dy/dx + 4 + dy/dx + 2 = 8  

=> 5·dy/dx + 6 = 8  

=> 5·dy/dx = 2  

=> dy/dx = 2/5

Explanation

[−73^7]^6 = 7x

First, simplify the left-hand side:  

= (−73^7)^6  

= −73^(7×6) = −73^42  

So:

−73^42 = 7x  

Now solve for x:  

x = (−73^42) / 7

Explanation