Explanation

Arthur Cayley, a British mathematician, introduced the modern concept of matrices in 1858.

He developed matrix algebra and its applications in linear transformations.

Explanation

a² - b² = (a - b)(a + b)

a⁴ - b⁴ = (a² - b²)(a² + b²)

= (a - b)(a + b)(a² + b²)

The LCM is (a - b)(a + b)(a² + b²) = (a² - b²)(a² + b²)

= a⁴ - b⁴

Explanation

Given 12 = PQ, we can write P = 12/Q.

For proportion:

4 : Q = P : 3

4/ Q = P / 3

4 × 3 = P × Q

12 = PQ

This matches the given equation.

The answer is 4 : q = p : 3.

Explanation

Loga (m × n) = Loga m + Loga n.

This is a logarithmic property: the log of a product is the sum of the logs.

Explanation

(√x + √y) (√x - √y)

= (√x)² - (√y)²

= x - y

Explanation

x² - xy + y²

= (-2)² - (-2)(2) + (2)²

= 4 + 4 + 4

= 12

Explanation

(3a - 2b)² = (3a)² - 2(3a)(2b) + (2b)²

= 9a² - 12ab + 4b²

Explanation

The expression x² + 7x + 3 has three terms.

A polynomial with three terms is called a trinomial.

Explanation

a² - b² = (a - b)(a + b).

Explanation

a + b = 2 ... (1)

a - b = 2 ... (2)

Adding (1) and (2):

2a = 4

a = 2

Put a = 2 in (1):

2 + b = 2

b = 0

Now, a² + b² = (2)² + (0)² = 4 + 0 = 4.