Problem 35 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\sqrt{50}-\sqrt{72}$$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(\sqrt{50} - \sqrt{72}\). Answer: The simplified expression is \(-1\sqrt{2}\).

Step 1: Identify perfect square factors

In order to simplify the given expression, look for perfect square factors for each term. The largest perfect square factor of 50 is 25, since 25 * 2 = 50. The largest perfect square factor of 72 is 36, since 36 * 2 = 72. Now that we've identified the perfect square factors, let's proceed to the next step.

Step 2: Apply the square root property

Use the property, \(\sqrt{a \cdot b} = \sqrt{a}\cdot \sqrt{b}\), to separate the square root terms: $$\sqrt{50} - \sqrt{72} = \sqrt{25 \cdot 2} - \sqrt{36 \cdot 2}$$

Step 3: Simplify the square roots

Simplify the square root terms by evaluating the square roots of the perfect squares: $$\sqrt{25} \cdot \sqrt{2} - \sqrt{36} \cdot \sqrt{2}$$ $$= 5\sqrt{2} - 6\sqrt{2}$$

Step 4: Combine like terms

Now that both terms have a common square root factor, combine them by subtracting their coefficients: $$5\sqrt{2} - 6\sqrt{2} = -1\sqrt{2}$$ So the simplified expression is: $$-1\sqrt{2}$$

Key Concepts

Perfect Square Factors

Understanding perfect square factors is essential when simplifying radical 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 result of 4 multiplied by 4.

When faced with a radical expression like \( \sqrt{50} \) and \( \sqrt{72} \), identifying and extracting the largest perfect square factor simplifies the process. In \( \sqrt{50} \), we find that 25 is the largest perfect square factor because 25 times 2 equals 50. Similarly, for \( \sqrt{72} \) the factor is 36. These factors are crucial in reducing the radical to its simplest form.

Square Root Properties

Square root properties are rules that make working with radicals more manageable. One fundamental property is the multiplication property of square roots, which states that \( \sqrt{a \cdot b} = \sqrt{a}\cdot \sqrt{b} \). This property allows us to separate a square root of a product of numbers into the product of two separate square roots.

Applying this property to perfect square factors, as in the expressions \( \sqrt{25 \cdot 2} \) and \( \sqrt{36 \cdot 2} \), we can simplify them further by taking the square root of the perfect squares (25 and 36) separately from the non-perfect square (2), resulting in simplified expressions that can then be easily combined or compared.

Combine Like Terms

In algebra, combining like terms refers to the process of simplifying expressions by adding or subtracting terms that have the same variable or radical part. After applying square root properties and simplifying individual square roots, we often end up with radical expressions that include like terms.

For instance, in our previous example, the expressions \( 5\sqrt{2} \) and \( 6\sqrt{2} \) are like terms because they both contain the \( \sqrt{2} \) part. To combine them, simply adjust the coefficients—think of it as if you are dealing with 'x's or 'y's. You would just add or subtract the numerical coefficients while keeping the variable part intact. Similarly, in this case, we subtract the coefficients (5 and 6) while maintaining the \( \sqrt{2} \) part, resulting in \( -1\sqrt{2} \), which is the combined and simplified expression.