Problem 80 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{80 c} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4 \sqrt{5c} \).

Step 1: Identify Perfect Squares

Identify perfect squares within 80. Since 80 can be written as the product of 16 and 5, and 16 is a perfect square, we can rewrite the expression as \( \sqrt{16 \times 5 \times c} \).

Step 2: Simplify the Square Root

Apply the property of square roots that allows you to separate the product: \( \sqrt{16 \times 5 \times c} = \sqrt{16} \times \sqrt{5} \times \sqrt{c} \).

Step 3: Calculate the Perfect Square

Simplify \( \sqrt{16} \) which is equal to 4, as 16 is a perfect square: \( \sqrt{16} = 4 \).

Step 4: Express the Simplified Square Root

Substitute \( \sqrt{16} \) back into the expression, giving \( 4 \times \sqrt{5} \times \sqrt{c} \) which can also be written as \( 4 \sqrt{5c} \).

Key Concepts

Perfect Squares

A perfect square is a number that is the square of an integer. For example, numbers like 1, 4, 9, 16, 25 are perfect squares because they are the result of squaring the integers 1, 2, 3, 4, and 5, respectively. Recognizing perfect squares is crucial when simplifying square roots because they can be easily taken out of the square root sign.

In the case of the number 80, we look for its factors and identify any perfect squares. We find that 16 is a perfect square that divides 80 exactly (since 80 = 16 × 5). This means we can simplify \( \sqrt{80} \) by expressing it as \( \sqrt{16 \, \times 5} \). By taking \( \sqrt{16} \) out of the square root as 4, we simplify the expression significantly.

Properties of Square Roots

Understanding the properties of square roots is fundamental when dealing with radical expressions. Square roots have several properties that can help simplify complex expressions:

• \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) - This allows us to separate the square root of a product into the product of square roots.
• \( \sqrt{a^2} = a \) for \( a \geq 0 \) - This property defines that the square root of a perfect square returns to its base number.
• \( \sqrt{a/b} = \sqrt{a}/\sqrt{b} \) - This property helps to handle division under the square root.

Using these properties, we apply them to a number like 80c expressed in square roots: \( \sqrt{16 \times 5 \times c} = \sqrt{16} \times \sqrt{5} \times \sqrt{c} \). This step is crucial because it not only simplifies \( \sqrt{80} \) but also allows us to deal with each component separately.

Radical Expressions

Radical expressions involve square roots, cube roots, or any other higher-order roots, expressed as terms under radical signs. Simplifying radical expressions often involves breaking them down using perfect squares and applying square root properties.

A simplified radical expression should have no fractions under the radical sign, no radicals in the denominator, and the smallest possible number under the radical sign.

In our exercise, simplifying \( \sqrt{80c} \) involves rewriting \( 80 \) as \( 16 \times 5 \) and using the properties of square roots to separate it into \( \sqrt{16} \times \sqrt{5} \times \sqrt{c} \). The simplified radical expression becomes \( 4 \sqrt{5c} \), which is both neater and easier to understand.