Problem 86 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{200 x^{2} y} $$

Step-by-Step Solution

Answer

\(10x \sqrt{2y}\)

Step 1: Identify Prime Factors

Start by identifying the prime factors of the number under the square root. The number 200 can be broken into its prime factors as follows: 200 = 2^3 imes 5^2. This simplifies our expression under the square root.

Step 2: Separate the Radical

Rewrite the square root expression by separating the factors:\[\sqrt{200x^2y} = \sqrt{2^3 \times 5^2 \times x^2 \times y}\]This helps in simplifying each factor separately.

Step 3: Simplify Perfect Squares

Perfect squares can be pulled out of the square root as their own numbers. Since 5^2 and x^2 are perfect squares, they can be simplified:\[\sqrt{5^2} = 5, \quad \text{and} \quad \sqrt{x^2} = x.\]Write the simplified expression as:\[5x \sqrt{2^3 \cdot y}\]

Step 4: Further Simplify Remaining Terms

For the remaining terms inside the square root, \[2^3 \cdot y = 2^2 \cdot 2 \cdot y = 4 \cdot 2 \cdot y\]The square root of 4 is 2, so simplify:\[2 \sqrt{2y}\]Thus, the expression becomes:\[5x imes 2 \sqrt{2y} = 10x \sqrt{2y}.\]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its basic building blocks—prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To simplify square roots, understanding prime factorization is essential because it helps identify perfect squares that can be simplified out of the radical.

• Consider the number 200. Break it down into its prime factors by division: 200 can be expressed as \(2^3 \times 5^2\).
• This factorization helps in identifying parts of the number that can be simplified when they are perfect squares.

A solid grasp of prime factorization will assist in simplifying any square root by isolating those factors which can be easily removed or reduced.

Perfect Squares

Perfect squares are numbers that have a whole number as their square root. Identifying perfect squares is crucial when simplifying square roots because it allows portions of the expression to be taken out from under the radical.

• In the expression \(\sqrt{200x^2y}\), the numbers and variables inside the radical can be analyzed individually.
• Both \(5^2\) and \(x^2\) are perfect squares, meaning \(\sqrt{5^2} = 5\) and \(\sqrt{x^2} = x\).

This means that these parts can be simplified completely, resulting in a much simpler expression outside the square root sign.

Square Root Properties

Understanding the properties of square roots is vital for successful simplification. One of these properties is that the square root of a product is the product of the square roots.

• For an expression like \(\sqrt{200x^2y}\), separate the multipliers under the square root for easier simplification: \(\sqrt{200} \times \sqrt{x^2} \times \sqrt{y}\).
• This lets you evaluate each part independently, taking perfect squares out of the radical, as with \(5^2\) and \(x^2\).

Such properties assist in breaking down complex expressions into manageable parts, ultimately helping to further simplify the expression.

Expressions with Variables

Expressions involving variables can sometimes be daunting, especially under a square root. The approach is to consider each variable as you would a constant number.

• In our expression \(\sqrt{200x^2y}\), the goal is to treat \(x^2\) just like a regular number that is a perfect square.
• Simply simplify \(x^2\) to \(x\), taking it out of the square root with ease.
• As for \(y\), since it’s not a perfect square, it remains under the radical unless further information simplifies it.

Understanding that variables can sometimes act like numbers can help demystify such expressions and lead to faster simplification.