Question

5 the expression \\(\sqrt{150}\\) is equivalent to

1. \\(25\sqrt{6}\\)
2. \\(15\sqrt{10}\\)
3. \\(5\sqrt{6}\\)
4. \\(6\sqrt{5}\\)

6 when \\(5\sqrt{20}\\) is written in simplest radical form, the result is \\(k\sqrt{5}\\). what is the value of \\(k\\)?

1. 20
2. 10
3. 7
4. 4

7 what is \\(2\sqrt{45}\\) expressed in simplest radical form?

1. \\(3\sqrt{5}\\)
2. \\(5\sqrt{5}\\)
3. \\(6\sqrt{5}\\)
4. \\(18\sqrt{5}\\)

Explanation:

Question 5

Step1: Factor 150

We factor 150 into a product of a perfect square and another number. We know that \(150 = 25\times6\), where 25 is a perfect square.

Step2: Simplify the square root

Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)), we have \(\sqrt{150}=\sqrt{25\times6}=\sqrt{25}\times\sqrt{6}\).
Since \(\sqrt{25} = 5\), then \(\sqrt{150}=5\sqrt{6}\).

Step1: Simplify \(\sqrt{20}\)

First, factor 20. We get \(20 = 4\times5\), where 4 is a perfect square. Then \(\sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}\).

Step2: Multiply by 5

Now, we have \(5\sqrt{20}=5\times(2\sqrt{5})\). Using the associative property of multiplication, \(5\times2\times\sqrt{5}=10\sqrt{5}\).
Since the result is \(k\sqrt{5}\), then \(k = 10\).

Step1: Simplify \(\sqrt{45}\)

Factor 45: \(45=9\times5\), and 9 is a perfect square. So \(\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\).

Step2: Multiply by 2

Now, \(2\sqrt{45}=2\times(3\sqrt{5})\). Using the associative property of multiplication, \(2\times3\times\sqrt{5}=6\sqrt{5}\).

Answer:

1. \(5\sqrt{6}\)

Question 6