Problem 40 - College Algebra

Question

Simplify each expression. $$9 \sqrt{3 m^{2}}+\sqrt{27}$$

Step-by-Step Solution

Answer

Simplified expression: \((9m + 3)\sqrt{3}\).

Step 1: Simplify the First Square Root

The first part of the expression is \(9 \sqrt{3m^2}\). Since \(m^2\) is a perfect square, you can simplify it as \(m\). So, \(9 \sqrt{3m^2} = 9m\sqrt{3}\).

Step 2: Simplify the Second Square Root

The second part is \(\sqrt{27}\). We can simplify this by factoring \(27\) as \(9 \times 3\). Since \(9\) is a perfect square, \(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\).

Step 3: Combine the Simplified Expressions

Now, combine the simplified terms from Steps 1 and 2: \(9m\sqrt{3} + 3\sqrt{3}\). Since both terms have a \(\sqrt{3}\) in common, factor it out: \((9m+3)\sqrt{3}\).

Step 4: Final Simplification

Observe that there are no common factors in \(9m + 3\) other than \(\sqrt{3}\) already factored out. So the simplified expression is \((9m + 3)\sqrt{3}\).

Key Concepts

Square roots

Square roots are an important part of algebra. They can seem tricky at first, but they're easier once you understand the basics. A square root is a number that, when multiplied by itself, gives you the original number.
For example, the square root of 9 is 3 because 3 times 3 equals 9. It's often represented with the radical symbol \( \sqrt{} \). When you see \( \sqrt{27} \), it means "what number times itself equals 27?"
Squaring brings us back to earlier math lessons about multiplication. Remember that square roots can also involve variables, like \( \sqrt{3m^2} \). This means you're going to need to factor out perfect squares to simplify, which leads us to our next topic.

Perfect squares

A perfect square is a number that is the square of an integer. Think of it as a number that has been multiplied by itself once.
For instance:

• 1 (because 1 x 1 = 1)
• 4 (because 2 x 2 = 4)
• 9 (because 3 x 3 = 9)

Understanding perfect squares helps us simplify square roots. When you see \( \sqrt{27} \), recognize that 27 can be broken down into \( 9 \times 3 \).
Since 9 is a perfect square, \( \sqrt{27} = \sqrt{9} \times \sqrt{3} = 3 \times \sqrt{3} = 3\sqrt{3} \). Recognizing perfect squares in expressions makes simplification much easier.

Factoring

Factoring is the process of breaking down a number or expression into a product of other numbers or expressions. It's like finding what can be multiplied together to get the original number.
In our example, the number 27 was factored into 9 and 3 because 9 times 3 equals 27. When simplifying square roots, factoring is essential because it can help make the problem easier by isolating perfect squares.
With expressions like \( \sqrt{3m^2} \), understand that \( m^2 \) is a perfect square. This makes simplifying straightforward, turning \( 3m^2 \) into \( m \cdot m \).

• By recognizing perfect squares through factoring, the simplification process becomes much more manageable.
• You're often left with a more elegant and understandable expression.

Common factors

Common factors refer to numbers or expressions that divide two or more terms evenly. Finding common factors is crucial for simplification and can make your final expressions as neat as possible.
When combining terms like \( 9m\sqrt{3} + 3\sqrt{3} \), you notice both parts have a \( \sqrt{3} \). By factoring out \( \sqrt{3} \), you're left with \( (9m + 3)\sqrt{3} \).
In another simplification context, if you had something like \( 9m + 3 \), you'd look internally for additional common factors, separate from \( \sqrt{3} \).

• This understanding helps you tidy up and organize mathematical expressions.
• Ultimately, recognizing and factoring out common factors leads to cleaner solutions and immense clarity.