Question 1

What does it mean to simplify a square root?

Accepted Answer

Simplifying a square root means expressing it in its simplest radical form by factoring out perfect squares. For example, √8 = √(4 × 2) = √4 × √2 = 2√2. The goal is to write the square root as a coefficient times the square root of a smaller number that has no perfect square factors.

Question 2

How do you simplify square roots step by step?

Accepted Answer

To simplify √n: 1) Find the largest perfect square that divides n. 2) Factor n as (perfect square) × (remaining factor). 3) Use the property √(a × b) = √a × √b. 4) Simplify √(perfect square) to its integer value. 5) Write the result as coefficient√remaining. Example: √12 = √(4 × 3) = √4 × √3 = 2√3.

Question 3

What are perfect squares?

Accepted Answer

Perfect squares are numbers that are the square of integers: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, etc. These are crucial for simplifying square roots because √(perfect square) = integer.

Question 4

How do you find the largest perfect square factor?

Accepted Answer

To find the largest perfect square factor of a number: 1) Start with the largest integer whose square is less than or equal to the number. 2) Check if that square divides the number evenly. 3) If not, try the next smaller integer. 4) Continue until you find the largest perfect square that divides the number. This gives you the maximum simplification possible.

Question 5

What is the difference between radical form and decimal form?

Accepted Answer

Radical form keeps the square root symbol (e.g., 2√3 ≈ 3.464) and shows the exact mathematical relationship. Decimal form gives the numerical approximation (e.g., √12 ≈ 3.464). Radical form is preferred in algebra because it's exact, while decimal form is useful for practical calculations and comparisons.

Question 6

When is a square root already simplified?

Accepted Answer

A square root is already simplified when the number under the radical has no perfect square factors other than 1. This means the number is either prime or a product of distinct primes. Examples: √2, √3, √5, √6, √7, √10, √11, √13, √14, √15 are already simplified.

Question 7

How do you simplify square roots with variables?

Accepted Answer

For expressions like √(x²y³): 1) Factor out perfect squares from the exponents. 2) Use √(x²) = |x| and √(y²) = |y|. 3) Apply the property √(a × b) = √a × √b. Example: √(x²y³) = √(x²y²y) = √(x²y²) × √y = |x||y|√y = |xy|√y.

Question 8

What are the properties of square roots used in simplification?

Accepted Answer

Key properties: 1) √(a × b) = √a × √b (product property). 2) √(a/b) = √a/√b (quotient property). 3) √(a²) = |a| (square root of a square). 4) √(a²b) = |a|√b (factoring out perfect squares). These properties allow us to break down complex square roots into simpler components.

Question 9

How do you check if your simplification is correct?

Accepted Answer

To verify simplification: 1) Square both the original and simplified forms. 2) They should give the same result. 3) Check that the simplified form has no perfect square factors in the radical. Example: √8 = 2√2, so (2√2)² = 4 × 2 = 8 ✓. Also verify that 2 has no perfect square factors ✓.

Question 10

Why is simplifying square roots important in algebra?

Accepted Answer

Simplifying square roots is crucial because: 1) It makes expressions easier to work with in equations. 2) It reveals the mathematical structure of the number. 3) It's required for adding/subtracting like radicals. 4) It helps identify equivalent expressions. 5) It's essential for solving quadratic equations and working with the quadratic formula.

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