Problem 1 - Algebra 2

Question

Simplify. $$ \sqrt{56} $$

Step-by-Step Solution

Answer

The simplified form is \( 2\sqrt{14} \).

Step 1: Identify Perfect Square Factors

The first step in simplifying a square root is to find the largest perfect square factor of the number. Analyze the factors of 56 to find the largest perfect square. The factors of 56 include 1, 2, 4, 7, 8, 14, 28, and 56. Among these, the largest perfect square is 4.

Step 2: Rewrite the Square Root

Rewrite the square root expression by expressing 56 as a product of its factors, incorporating the largest perfect square, 4. This gives us: \( \sqrt{56} = \sqrt{4 \times 14} \).

Step 3: Simplify the Square Root

Next, apply the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). So, we have: \( \sqrt{4 \times 14} = \sqrt{4} \times \sqrt{14} \). Since \( \sqrt{4} = 2 \), the expression simplifies to: \( 2 \times \sqrt{14} \).

Step 4: Confirm Simplification

Verify there are no further perfect square factors in 14. Since 14 is the product of 2 and 7, neither of which are perfect squares, \( 2 \times \sqrt{14} \) is in its simplest form.

Key Concepts

Perfect Squares

In mathematics, a perfect square is a number that is the square of an integer. For example, numbers like 4, 9, 16, and 25 are all perfect squares. In terms of multiplication:

• 4 is a perfect square because it is equal to 2 × 2,
• 9 is a perfect square because it is equal to 3 × 3,
• 16 is a perfect square because it is equal to 4 × 4.

Perfect squares are useful when simplifying square roots because they make calculations easier. When simplifying, one identifies the largest perfect square factor within the number under the square root. For example, when simplifying \( \sqrt{56} \), the goal is to find if 56 contains any perfect squares. Among its factors (1, 2, 4, 7, 8, 14, 28, and 56), the largest perfect square is 4. Knowing this allows us to break down \( \sqrt{56} \) into a simpler form, leveraging the property of square roots for easier calculation.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that is not divisible by any other numbers except 1 and itself. Examples include 2, 3, 5, 7, 11, etc. Prime factorization aids in simplifying square roots, as it helps find perfect square factors easily. Consider the number 56. Its factors can be broken down into prime numbers:

• Start with dividing 56 by 2, the smallest prime number, resulting in 28.
• Divide 28 by 2 to get 14, and continue this process to get:
• 56 = 2 × 2 × 2 × 7 = \(2^3 \times 7\).

Through prime factorization, we identify 4 (\(2^2\)) as a perfect square factor of 56. This is crucial in breaking down the square root further as in \( \sqrt{56} = \sqrt{4 \times 14} \). Recognizing and using these factors make the simplification process more straightforward.

Square Root Properties

Square roots possess certain properties that make them easier to manipulate in mathematical expressions. Understanding these helps in simplifying expressions effectively. Some key properties include:

• \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \): This property means the square root of a product is the product of the square roots.
• \( \sqrt{a^2} = a \): The square root of a perfect square is always an integer, which is the original number used to form that perfect square.

These properties are applied when simplifying the square root of a number. Consider \( \sqrt{56} \), which can be written as \( \sqrt{4 \times 14} \). By applying the product property, this becomes \( \sqrt{4} \times \sqrt{14} \). Since \( \sqrt{4} = 2 \), it can be further simplified to \( 2 \times \sqrt{14} \). This simplification ensures no perfect square factors are left, confirming that the expression is in its simplest form.