What is the product of the following expression in simplest radical form?

Question

√50 x √45

Answer 1

To simplify the expression \( \sqrt{50} \times \sqrt{45} \), we start by using the property of square roots that states \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). Therefore, we can combine the square roots:

\[ \sqrt{50} \times \sqrt{45} = \sqrt{50 \times 45} \]

Next, we calculate \( 50 \times 45 \):

\[ 50 \times 45 = 2250 \]

Now we have:

\[ \sqrt{50 \times 45} = \sqrt{2250} \]

To simplify \( \sqrt{2250} \), we factor \( 2250 \) into its prime factors. First, we divide \( 2250 \) by \( 10 \):

\[ 2250 = 225 \times 10 \]

Next, we focus on simplifying \( 225 \):

\[ 225 = 15 \times 15 = 15^2 \]

Now we return to \( 10 \):

\[ 10 = 2 \times 5 \]

Thus, we can write the factorization of \( 2250 \):

\[ 2250 = 15^2 \times 2 \times 5 = 3^2 \times 5^2 \times 2 \]

This can be combined as follows:

\[ 2250 = 2 \times 3^2 \times 5^2 \]

Now we apply the square root:

\[ \sqrt{2250} = \sqrt{2 \times 3^2 \times 5^2} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{5^2} \]

Knowing that \( \sqrt{3^2} = 3 \) and \( \sqrt{5^2} = 5 \), we have:

\[ \sqrt{2250} = \sqrt{2} \times 3 \times 5 = 15 \sqrt{2} \]

Thus, the product of the original expression \( \sqrt{50} \times \sqrt{45} \) in simplest radical form is:

\[ \boxed{15\sqrt{2}} \]

Answer 2

thats not correct

Answer 3

Let's re-evaluate the expression \( \sqrt{50} \times \sqrt{45} \) step by step to ensure accuracy in our simplification.

Step 1: Combine the Square Roots

Using the property of square roots that allows us to combine them:

\[ \sqrt{50} \times \sqrt{45} = \sqrt{50 \times 45} \]

Step 2: Calculate \( 50 \times 45 \)

Calculating the product:

\[ 50 \times 45 = 2250 \]

Step 3: Simplify \( \sqrt{2250} \)

Next, we need to simplify \( \sqrt{2250} \). Let's factor \( 2250 \).

Finding the prime factorization of \( 2250 \):

1. Start by dividing by \( 10 \): \[ 2250 = 225 \times 10 \] \[ 10 = 2 \times 5 \]

2. Now factor \( 225 \): \[ 225 = 15 \times 15 = 15^2 \] \[ 15 = 3 \times 5 \quad \text{so,} \quad 225 = (3 \times 5)^2 = 3^2 \times 5^2 \]

Thus, \[ 2250 = 2 \times 3^2 \times 5^2 \]

Step 4: Taking the Square Root

Using the square root property:

\[ \sqrt{2250} = \sqrt{2 \times 3^2 \times 5^2} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{5^2} \]

Calculating the roots:

\[ \sqrt{3^2} = 3 \quad \text{and} \quad \sqrt{5^2} = 5 \]

Therefore:

\[ \sqrt{2250} = \sqrt{2} \times 3 \times 5 = 15 \sqrt{2} \]

Conclusion

The product \( \sqrt{50} \times \sqrt{45} = \sqrt{2250} = 15\sqrt{2} \) was simplified correctly.

Thus, the correct answer is:

\[ \boxed{15\sqrt{2}} \]

If you believe there is still an error, please clarify or provide additional details.