Explanation

Deduction –  Reasoning from general principles to a specific conclusion

 This is a classic example of deductive reasoning, specifically a syllogism.

It demonstrates how a conclusion can be logically derived from two premises

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

A complex cube root of unity is a complex number $omega$ such that:

$$omega^3 = 1,quad omega neq 1$$

So:

$$$$ω3=1omega^3 = 1

Explanation

Step 1: Rule for divisibility by 5

A number is divisible by 5 if and only if it ends in 0 or 5.
But 0 is not among the given digits, so the last digit must be 5.

Step 2: Fix the last digit as 5

So, we fix 5 as the last digit.

Now we must choose the remaining 4 digits (from the remaining digits: 2, 3, 7, 9)
These will go in the first 4 places.

• Number of ways to arrange 4 digits out of 4 = 4! = 24

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

Let number of boys = x
Then girls = x + 7
Total = x + x + 7 = 27
2x + 7 = 27 → 2x = 20 → x = 10
Girls = 10 + 7 = 17