Explanation

= 3¹⁴ ÷ 9² = 3^x

= 9² = (3²)² = 3⁴

= 3¹⁴ ÷ 3⁴ = 3¹⁴-⁴ = 3¹⁰

= x = 10

Explanation

x / √128 = √162 / x

x² = √128 × √162

x² = √(128 × 162)

x² = √20736

x² = 144

x = 12

Explanation

Let c contribution be x then :

b = 1/4x

a = 2/3 × 1/4x = 1/6x

1/4x + 1/6x + x = 1360

6x + 4x + 24x = 1360 × 24

34x = 32640

x = 960

Contribution of b is :

1/4 × 960 = 240

Explanation

The additive inverse of a matrix A is the matrix which when added to A results in the zero matrix.  

So, additive inverse of matrix A = -A (negate each element).

Given matrix A =  

[ -7  8 ]  

[  6  3 ]

Additive inverse =  

[ 7  -8 ]  

[ -6  -3 ]

Explanation

(a² − b²) is a standard algebraic identity known as the difference of squares.

(a² − b²) = (a − b)(a + b)

Explanation

cosθ = b/h

- cosθ = adjacent/hypotenuse

- Here, b = base (adjacent), h = hypotenuse

Explanation

Transpose of a matrix means converting rows into columns.  

Original matrix:  

[ 2 9 ]  

[ 3 4 ]  

Transpose:  

[ 2 3 ]  

[ 9 4 ]

Explanation

Given the operation:

c * d = (c - d) / c

For 12 * 3:

12 * 3 = (12 - 3) / 12

= 9 / 12

= 3 / 4

Explanation

Given equations:

p + 2q = 14 ... (1)

3p + q = 12 ... (2)

Multiply (2) by 2:

6p + 2q = 24 ... (3)

Subtract (1) from (3):

(6p - p) + (2q - 2q) = 24 - 14

5p = 10

p = 2

Explanation

Given equations:

m - n = 5 ... (1)

2m + 3n = 15 ... (2)

From (1), m = n + 5

Substitute m in (2):

2(n + 5) + 3n = 15

2n + 10 + 3n = 15

5n + 10 = 15

5n = 5

n = 1

Now, find m:

m = n + 5

m = 1 + 5

m = 6

m + n = 6 + 1 = 7