What is the definition of a square root?

The square root of a number is a value that, when multiplied by itself, equals the original number. 
For Example, 
√25 = 5 
because, 5 × 5 = 25.

What is the formula for the square of a binomial

(a + b)²?

(a + b)² = a² + 2ab + b². 
For example, 
(2 + 3)² = 2² + 2(2)(3) + 3²
 = 4 + 12 + 9 
= 25

How do you simplify the expression √(36x²)?

To simplify √(36x²), recognize that 36 is 6² and x² is x². 
Thus,√(36x²) = √(6²) * √(x²)

= 6x.

Solve for x: √(x + 9) = 7 (Hint: Square both sides to eliminate the squareroot)

Square both sides: 
(√(x + 9))² = 7², 
x + 9 = 49
Subtract 9 from both sides,we get

x = 40

The square root of 27225 is?

Square root of 27225

Using factorization:

27225 = 3 × 3 × 5 × 5 × 11 × 11

∴ Square root will be 
= 3 × 5 × 11 = 165

∴ √27225 = 165

What is the cube root of a number?

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. 
For Example, 
∛8 = 2 
because 2 × 2 × 2 = 8.

How do you simplify the expression ∛(27x⁶)?

To simplify ∛(27x⁶), recognize that 27 is 3³ and x⁶ is (x²)³. 
Thus, 
∛(27x⁶) = ∛(3³) * ∛((x²)³) 
= 3x².

Find the value of 
70³ + 20³ - 90³?

We know that 
a³ + b³ + c³ = 3abc 
if a + b + c = 0

70³ + 20³ - 90³
= 70³ + 20³ + (-90)³
= 3 × 70 × 20 × (-90) 
[∴ 70 + 20 + (-90) = 0]
= -378000

Solve for x: 
∛(2x + 1) = 3. 

(Hint: Cube both sides to eliminate the cube root)

Cube both sides: 
(∛(2x + 1))³ = 3³, 
which simplifies to 
2x + 1 = 27. 
Subtract 1 from both sides, 
2x = 26, 
so x = 13.

What is the formula for the difference of squares?

The difference of squares is given by 
a² - b² = (a - b)(a + b). 
For example, 
4² - 3² = (4-3)(4+3) 
= 1 × 7 
= 7

How can you approximate the square root of a non-perfect square like 20?

To approximate √20, 
Note that it falls between √16 = (4) and √25=(5). 
A rough estimate would be around 4.5