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 square root.

Square both sides: (√(x + 9))² = 7², which simplifies to x + 9 = 49. Subtract 9 from both sides: x = 40.

What is the relationship between the square root and exponents?

The square root of a number x can be expressed as x^(1/2). For example, √(16) = 16^(1/2) = 4.

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.

If x = 3, what is the value of 2√x + 3∛x? Hint: Calculate each root separately.

First, calculate √3 and ∛3. Thus, 2√3 + 3∛3 = 2(1.732) + 3(1.442) ≈ 3.464 + 4.326 = 7.790.

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, 9 - 16 = (3 - 4)(3 + 4) = -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.