Explanation

To expand the expression:

(√3 - √2)²

= (√3)² - 2(√3)(√2) + (√2)²

= 3 - 2√6 + 2

= 5 - 2√6

Explanation

Explanation

 Let's solve the equation:

(3/2)x - 5/2 = 1/2

Add 5/2 to both sides:

(3/2)x = 1/2 + 5/2

(3/2)x = 6/2

(3/2)x = 3

Multiply both sides by 2/3:

x = 3 × (2/3)

x = 2

Explanation

Let's solve for x:

x/2 + x/3 + x/6 = 10

To add these fractions, find a common denominator (6):

3x/6 + 2x/6 + x/6 = 10

6x/6 = 10

x = 10

Explanation

Let's simplify the expression:

√25^(3x) = √(5²)^(3x) = √(5^(6x)) = 5^(3x)

Now, divide by 5^(6x-2):

5^(3x) / 5^(6x-2) = 5^(3x - 6x + 2) = 5^(-3x + 2)

For this to equal one of the given options (5, 10, 15), let's see if -3x + 2 = 1 (for 5¹ = 5):

-3x + 2 = 1

-3x = -1

x = 1/3

Then, 5^(-3(1/3) + 2) = 5¹ = 5

Explanation

The expression √8-√2 simplifies to √2.

Explanation

A group generated by one of its elements is called a cyclic group.

This means that the group can be generated by repeatedly applying the group operation to a single element, called the generator.

In other words, all elements of the group can be expressed as powers of the generator.

Explanation

If AB = BA, it means the matrices commute under multiplication.

So, they are called multiplicative commutative (or simply commutative) matrices.

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

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.