Problem 111 Solve each problem. Find the e... [FREE SOLUTION] (2024)

Short Answer

Multiplying Square Roots

When you multiply square roots, you can combine the values under the root if they are both non-negative. For example, to multiply \( \sqrt{6} \) and \( \sqrt{3} \), you follow this rule: \[ \sqrt{6} \times \sqrt{3} = \sqrt{6 \times 3} \]. The value under the square root becomes \( 18 \). This property is called the product property of square roots. It makes calculations simpler and is highly useful in geometry when working with lengths and areas.

Simplifying Square Roots

After multiplying the values under the square roots, you often need to simplify the result. Let's take our example further, where we ended up with \( \sqrt{18} \). First, find the prime factors: \( 18 = 9 \times 2 \). Knowing that \( 9 = 3^2 \), we write: \[ \sqrt{18} = \sqrt{3^2 \times 2} \].
We separate the square and use the rule that \( \sqrt{a^2} = a \): \[ \sqrt{18} = \sqrt{3^2} \times \sqrt{2} = 3 \sqrt{2} \]. Always look for perfect squares inside the root to simplify your expression. Simplification is important for exact answers, especially in measurements.

Geometry Formulas

To calculate areas, perimeters, and other geometric quantities, you need to know and apply the correct formulas. The area of a rectangle is calculated using: \[ \text{Area} = \text{Length} \times \text{Width} \].
For our exercise, the area formula was \( \sqrt{6} \times \sqrt{3} \), which simplifies to: \[ \text{Area} = 3 \sqrt{2} \: \text{ft}^2 \]. Understanding and using these formulas correctly allows you to solve various geometry problems accurately. Make sure you practice these formulas to become proficient in solving geometry exercises efficiently.