Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{3} \cdot \sqrt{27} $$

Step-by-Step Solution

Answer

The product simplifies to 9.

Step 1: Apply the Property of Radicals

The property of radicals states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Using this property, we can rewrite the given expression \( \sqrt{3} \cdot \sqrt{27} \) as one single square root: \( \sqrt{3 \cdot 27} \).

Step 2: Calculate the Product Inside the Radical

Calculate the product of the numbers inside the radical: \( 3 \cdot 27 = 81 \). So the expression simplifies to \( \sqrt{81} \).

Step 3: Simplify the Square Root

Find the square root of 81. Since \( 81 = 9^2 \), we have \( \sqrt{81} = 9 \).

Key Concepts

Understanding the Property of Radicals

When dealing with radicals, one of the essential properties to know is the "property of radicals". This property is incredibly useful when simplifying radical expressions, particularly when you are multiplying two or more radicals together. This property states that: \[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \] This means if you have two square roots being multiplied, you can combine them into a single square root, containing the product of the two values. Using this property can help you simplify expressions rapidly,removing any complexity that might arise from dealing with multiple radicals.

• Combines multiple radical expressions into one.
• Helps in simplifying calculations.
• Boosts the efficiency of solving problems involving radicals.

In our exercise, we applied this property by rewriting \( \sqrt{3} \cdot \sqrt{27} \) as \( \sqrt{3 \cdot 27} \). Exploiting this property helps pave the way for further simplification.

Exploring Square Roots

The square root is one of the most recognized mathematical operations involving radicals. When you calculate a square root, you are essentially finding a number, which, when multiplied by itself, gives the original number under the square root symbol (radicand). For example, \( \sqrt{81} \) is asking, "What number multiplied by itself equals 81?" The answer, in this case, is 9 because \( 9 \times 9 = 81 \).Square roots can sometimes lead to a simpler form when the radicand is a perfect square. Some tips include:

• If the radicand is a perfect square (like 81), the expression simplifies nicely.
• Recall the "squares" of commonly known numbers to speed up simplification.

In our example, after applying the property of radicals, we calculated the product to get \( \sqrt{81} \), which simplified to a neat integer: 9. This is the beauty of square roots in action.

Mastering the Art of Multiplying Radicals

Multiplying radicals is another critical operation that many students encounter in algebra. It might sound complicated at first, but with a strong grasp of the property of radicals, it becomes much easier. When multiplying radicals such as \( \sqrt{a} \) and \( \sqrt{b} \), following the property of radicals transforms it into just finding \( \sqrt{a \times b} \).Here's a step-by-step approach:

• Use the property of radicals to combine the square roots.
• Calculate the product under a single radicand.
• Simplify further if possible, using square root properties.

In our particular scenario with \( \sqrt{3} \cdot \sqrt{27} \), the process was straightforward.Combining them gave \( \sqrt{81} \), which seamlessly resolved to 9. Mastering this art not only simplifies solutions but enhances problem-solving agility in algebra.