Problem 27 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{200} $$

Step-by-Step Solution

Answer

The simplified expression for \( \sqrt{200} \) is \( 2\sqrt{50} \)

Step 1: Identify the Factors

First, we need to identify the factors of the number 200 that are perfect squares. Examining the factors, we can see that 200 can be written as \(4 \times 50\). Here, 4 is a perfect square.

Step 2: Use the Property of Square Roots

We can think of \( \sqrt{200} \) as \( \sqrt{4 \times 50} \). From the properties of square roots, we know that \( \sqrt{a \times b} \) can become \( \sqrt{a} \times \sqrt{b} \). So, \(\sqrt{200}\) becomes \( \sqrt{4} \times \sqrt{50} \).

Step 3: Solve the Square Roots

Calculating both square roots, \( \sqrt{4} \) equals 2 and \( \sqrt{50} \) does not simplify further. Thus, we reach the simplified expression \( 2\sqrt{50} \)

Key Concepts

Perfect Squares

Understanding perfect squares is fundamental when simplifying square root expressions. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it is the product of 4 times 4, usually written as \(4^2\).
Perfect squares are critical in simplification because the square root of a perfect square is always an integer. This means that when you come across a perfect square under a square root symbol, you can easily simplify it to its integer root. In the context of our original exercise, \(4\) is a perfect square since \(2 \times 2 = 4\), which allows for simpler solutions.

Properties of Square Roots

When simplifying square roots, it's essential to understand their properties. The square root function essentially asks the question: 'What number, when multiplied by itself, gives the original number?' One of the main properties used in simplification is that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This property allows us to break down complex square roots into simpler components that can be easier to evaluate.
In the exercise, this property is applied by breaking down \(\sqrt{200}\) into \(\sqrt{4} \times \sqrt{50}\), simplifying the root of the perfect square (4), and keeping the second component (\(\sqrt{50}\)) as it is not a perfect square.

Factorization in Algebra

Factorization is a powerful tool in algebra that involves breaking down numbers or expressions into factors that can be multiplied together to give the original number or expression. In the context of square roots, factorization helps find the perfect squares within a number, thereby simplifying the square root.
For our exercise, we used factorization to decompose 200 into \(4 \times 50\). With further factorization, if necessary, you can break down the number under the square root even more, revealing additional perfect squares that can simplify the expression further. Always look for the largest perfect square within a number to simplify square roots most efficiently.