Question

Simplify each expression. Assume that all variables are positive when they appear. $$(5 \sqrt{8})(-3 \sqrt{3})$$

Short answer

-30 \sqrt{6}

Step-by-step solution

- Simplify the Square Roots

Simplify each square root in the expression. Notice that \( \sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2} \) and \( -3 \sqrt{3} \) stays as it is.

- Write in Simplified Form

Rewrite the expression with simplified square roots: \( 5 \sqrt{8} = 5 \cdot 2 \sqrt{2} \) so the expression becomes \( (10 \sqrt{2})(-3 \sqrt{3}) \).

- Multiply the Constants

Multiply the constants (numbers) together: \( 10 \cdot -3 = -30 \).

- Multiply the Square Roots

Multiply the square roots together: \( \sqrt{2} \cdot \sqrt{3} = \sqrt{6} \).

- Combine the Results

Combine the results from steps 3 and 4: \( -30 \sqrt{6} \).

Key concepts

Simplifying Square Roots

When working with square roots, simplifying them is an essential skill.
The process involves breaking down the number inside the square root into its prime factors.
Next, pair the prime numbers and bring them out of the square root as one number.
For example, let's look at how we can simplify \( \sqrt{8} \) :
\( \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{2^2 \cdot 2} = 2 \sqrt{2} \).
Here, 4 was broken down into \( 2^2 \), and one 2 came out of the square root.
Sometimes, the square root might remain in its original form if it cannot be simplified further.
For instance, \( \sqrt{3} \) is already in its simplest form because 3 is a prime number.
Once each square root is simplified, the expression becomes easier to manage.

Multiplying Square Roots

After simplifying square roots, you often need to multiply them.
When multiplying square roots, use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \).
For example, multiplying \( \sqrt{2} \) and \( \sqrt{3} \) would yield: \( \sqrt{2} \cdot \sqrt{3} = \sqrt{6} \).
This property helps simplify complex expressions involving square roots.
Remember, the final result should be checked to see if it can be further simplified.
In our exercise, the simplified expression involved multiplying \( 10 \sqrt{2} \cdot -3 \sqrt{3} \).
Breaking it down, this becomes multiplying \( \sqrt{2} \cdot \sqrt{3} = \sqrt{6} \).
We then multiply this result with any constants (numbers) outside the square roots.

Multiplying Constants

When simplifying expressions with square roots, you often have constants (numbers) outside the square roots.
Combining these constants correctly is essential for an accurate answer.
In our example, we started with \( 5 \sqrt{8} \) and \( -3 \sqrt{3} \).
After simplifying the square roots, it became \( 10 \sqrt{2} \cdot -3 \sqrt{3} \).
Here, we separated the constants (10 and -3) from the square roots and multiplied them as regular numbers:
\( 10 \cdot -3 = -30 \).
Now, we combine this result with our previously obtained simplified square root:
\( -30 \cdot \sqrt{6} = -30 \sqrt{6} \).
Therefore, knowing how to correctly handle and multiply constants helps in effectively simplifying the final expression.