Problem 237 - Elementary Algebra

Question

In the following exercises, simplify. $$ (5 \sqrt{2})(3 \sqrt{6}) $$

Step-by-Step Solution

Answer

\( 30 \sqrt{3} \)

Step 1 - Multiply the Coefficients

First, multiply the coefficients (numbers outside the square roots): \( 5 \times 3 \). This gives the product: 15.

Step 2 - Multiply the Radicands

Next, multiply the numbers inside the square roots (radicands): \( \sqrt{2} \times \sqrt{6} \). This simplifies to \( \sqrt{12} \) because \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).

Step 3 - Simplify the Square Root

Now, simplify \( \sqrt{12} \). Factor 12 into its prime factors: \( 12 = 2 \times 2 \times 3 \). This simplifies to \( 2 \sqrt{3} \).

Step 4 - Combine the Results

Finally, multiply the coefficient from Step 1 with the simplified square root from Step 3: \( 15 \times 2 \sqrt{3} \). This gives: \( 30 \sqrt{3} \).

Key Concepts

multiplying coefficients

When dealing with radical expressions, one of the first steps is to multiply the coefficients. Coefficients are the numbers in front of the square root symbol. In our exercise, we had the expression \((5 \sqrt{2})(3 \sqrt{6})\). The numbers 5 and 3 are the coefficients.

To simplify, we start by multiplying these coefficients together:

• 5 × 3 = 15

This gives us a new expression:
\(15 \sqrt{2} \cdot \sqrt{6}\)

Always ensure to multiply coefficients together before moving on to simplify the radicands.

multiplying radicands

Next, we tackle the radicands. Radicands are the numbers inside the square root symbols. In our example:
\((5 \sqrt{2})(3 \sqrt{6})\), the numbers 2 and 6 are the radicands.

To combine these, we multiply them together:

• \( \sqrt{2} \times \sqrt{6} = \sqrt{2 \times 6}\)

This simplifies to:

\( \sqrt{12} \)

This is because the property of square roots allows us to combine them as long as they are under the same root symbol.

Now we have the expression
\( 15 \sqrt{12}\)

It's important to always perform the radicand multiplication after combining the coefficients.

simplifying square roots

The final important step in simplifying radical expressions is simplifying the resulting square roots. In our problem, we now have
\( 15 \sqrt{12} \)

We need to simplify \( \sqrt{12} \). First, we factor 12 into prime factors:

• 12 = 2 × 2 × 3

Using properties of square roots, we can pair the factors:

• \( \sqrt{12} = \sqrt{2 \times 2 \times 3} \)
• \( \sqrt{2 \times 2} \times \sqrt{3} \)
• \( 2 \sqrt{3} \)

So now our simplified expression is
\( 15 \times 2 \sqrt{3} = 30 \sqrt{3} \)

Combining the coefficient and simplified square root gives the final answer. Make sure to simplify square roots by factoring into prime factors to find pairs.