1. simplify the radical expression.

√45

5√3
3√5
3√15**
15

2. Simplify the radical expression
√180x^2

90x**
6x√5
5x√6
6√5x^2

3. simplify he radical expression
√250h^4 k^5

hk√125
5√10h^4 k^5
5h^2 k^2√10k**
25hk√10k

1. Why did the mathematician go to the doctor? Because he had too many square roots! The simplified radical expression for √45 is 3√15. Keep those roots in check!

2. Did you hear about the radical expression that went to the gym? It got all pumped up and simplified to 90x! So the simplified form of √180x^2 is 90x.

3. Hold on tight, because this radical expression is about to take you on a wild ride! The simplified form of √250h^4 k^5 is 5h^2 k^2√10k. It's like a rollercoaster of simplification!

1. To simplify the radical expression √45, we can factorize 45 into its prime factors: 45 = 3 * 3 * 5. Since 3 is a perfect square, we can take it out of the square root: 3√5. Therefore, the simplified radical expression is 3√5.

2. To simplify the radical expression √180x^2, we can factorize 180 into its prime factors: 180 = 2 * 2 * 3 * 3 * 5. Since 2 and 3 are perfect squares, we can take them out of the square root. Also, since x^2 is a perfect square, we can take it out of the square root as well. This gives us 6x√5. Therefore, the simplified radical expression is 6x√5.

3. To simplify the radical expression √250h^4k^5, we can factorize 250 into its prime factors: 250 = 2 * 5 * 5 * 5. Since 5 is a perfect square, we can take it out of the square root. Also, since h^4 is an even power, we can take h^2 out of the square root as well. However, k^5 has an odd power, so we have to leave it inside the square root. This gives us 5h^2k^2√10k. Therefore, the simplified radical expression is 5h^2k^2√10k.

To simplify radical expressions, you need to find the largest perfect square factor in the radicand and simplify it. Here's how you can simplify each of the given radical expressions:

1. √45

To simplify √45, you need to find the largest perfect square factor of 45. In this case, it is 9 (3^2). You can rewrite 45 as 9 * 5. Then, take the square root of the perfect square factor:

√45 = √(9 * 5) = √9 * √5 = 3√5

Therefore, the simplified radical expression is 3√5.

2. √180x^2

To simplify √180x^2, first find the largest perfect square factor in 180, which is 9 (3^2). Rewrite 180 as 9 * 20. Simplify the square root of the perfect square factor:

√180x^2 = √(9 * 20) * √x^2 = √9 * √20 * x = 3√20 * x

You can simplify further by finding the largest perfect square factor in 20, which is 4 (2^2):

3√20 * x = 3 * √(4 * 5) * x = 3 * √4 * √5 * x = 3 * 2 * √5 * x = 6√5 * x

Hence, the simplified radical expression is 6√5 * x.

3. √250h^4 k^5

To simplify √250h^4 k^5, first find the largest perfect square factor in 250, which is 25 (5^2). Rewrite 250 as 25 * 10. Simplify the square root of the perfect square factor:

√250h^4 k^5 = √(25 * 10) * √h^4 k^5 = √25 * √10 * h^2 k^2 * √k

Simplify further by finding the largest perfect square factor in h^4, which is h^2:

√25 * √10 * h^2 k^2 * √k = 5 * √10 * h^2 k^2 * √k = 5h^2 k^2 * √10k

Therefore, the simplified radical expression is 5h^2 k^2 * √10k.

√45 = √(9*5) = √9√5 = 3√5

to factor out the 3, you need to factor out √9. You just factored out √3.
√45 = √3√15

√180x^2 = √(36*5x^2) = 6x√5

√250h^4 k^5 = √(25*10h^4 k^4*k) = 5h^2k^2√(10k)
Interesting that you got this one right, though it is a bit harder than the ones you missed ...