Problem 10 - 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. $$ \sqrt{21} \cdot \sqrt{\frac{4}{3}} $$

Step-by-Step Solution

Answer

The simplest form is \(2\sqrt{7}\).

Step 1: Identify the Radicals

The problem involves two radicals: \( \sqrt{21} \) and \( \sqrt{\frac{4}{3}} \). The task is to express their product in simplest form.

Step 2: Use the Property of Radicals

Recall that the product of two square roots can be combined into a single square root: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Apply this property to rewrite the expression as \( \sqrt{21} \cdot \sqrt{\frac{4}{3}} = \sqrt{21 \cdot \frac{4}{3}} \).

Step 3: Simplify Inside the Square Root

Calculate the expression inside the square root: \( 21 \cdot \frac{4}{3} = \frac{21 \cdot 4}{3} \). Simplify by cancelling common factors: \( \frac{21}{3} = 7 \). Thus, we have \( \frac{21 \cdot 4}{3} = 7 \cdot 4 = 28 \).

Step 4: Take the Square Root

The simplified expression inside the square root is \( 28 \). The square root of 28 can be simplified by breaking it into \( \sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7} \).

Step 5: Final Step: State the Simplest Form

The simplest form of the product \( \sqrt{21} \cdot \sqrt{\frac{4}{3}} \) is \( 2\sqrt{7} \).

Key Concepts

Understanding Square Roots

The square root is a mathematical operation that helps us find a number which, when multiplied by itself, gives us the original number. Think of it as the opposite of squaring a number.
For example, the square root of 9 is 3 because when you multiply 3 by 3, you get 9.

• Represented as \( \sqrt{} \).
• Its purpose is to find the side length of a square if you know the area.

Square roots are crucial as they allow us to work backwards when we deal with perfect squares or to simplify radical expressions into more manageable forms.

Exploring Radical Expressions

Radical expressions are numbers that contain roots, such as square roots. These can have variables or numbers inside the radical sign \( \sqrt{} \). Handling these expressions often involves simplifying the number under the radical (the radicand).
To simplify radical expressions, look for perfect squares in the radicand:

• Check if numbers can be broken down into factors that include squares.
• Simplify by separating the perfect square factor out of the radicand.

For example, \( \sqrt{28} \) becomes \( 2\sqrt{7} \) because 28 breaks into \( 4 \times 7 \), where \( 4 \) is a perfect square. Radicals can also be handled with algebraic expressions and variables, where similar rules apply to simplify accordingly.

The Product of Two Radicals

Understanding how to multiply radicals is key when dealing with expressions like our problem \( \sqrt{21} \cdot \sqrt{\frac{4}{3}} \). The rule to remember here is that the product of two radicals can be combined into a single radical, as long as both are square roots.
You can simplify expressions by combining under one radical:

• Use the equation \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
• Simplify inside the radical representation before further reductions.

In our exercise, starting with \( \sqrt{21} \cdot \sqrt{\frac{4}{3}} \) becomes \( \sqrt{21 \cdot \frac{4}{3}} \). This streamlines the calculation process and allows you to work more efficiently with numbers that aren’t perfect squares or appearing as fractions.