Question

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$\sqrt{40}$ $\sqrt{44}$ $\sqrt{45}$ $\sqrt{54}$ $\sqrt{56}$ $\sqrt{60}$ $\sqrt{75}$ $\sqrt{80}$ $\sqrt{90}$ $\sqrt{108}$ $\sqrt{125}$ $\sqrt{150}$ $\sqrt{x^3}$ $\sqrt{x^7}$ $\sqrt{x^9}$

Answer:

To simplify these square roots, we use the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (where \(a\geq0\) and \(b\geq0\)) and identify perfect square factors. Let's simplify a few examples:

1. Simplify \(\boldsymbol{\sqrt{40}}\)

Step 1: Factor 40

Find the largest perfect square factor of 40. The factors of 40 are \(1, 2, 4, 5, 8, 10, 20, 40\). The largest perfect square is 4. So, \(40 = 4\times10\).

Step 2: Apply the square root property

\(\sqrt{40}=\sqrt{4\times10}=\sqrt{4}\cdot\sqrt{10}\)
Since \(\sqrt{4} = 2\), we have \(\sqrt{40}=2\sqrt{10}\).

2. Simplify \(\boldsymbol{\sqrt{44}}\)

Step 1: Factor 44

The largest perfect square factor of 44 is 4 (since \(44 = 4\times11\)).

Step 2: Apply the square root property

\(\sqrt{44}=\sqrt{4\times11}=\sqrt{4}\cdot\sqrt{11}=2\sqrt{11}\)

3. Simplify \(\boldsymbol{\sqrt{45}}\)

Step 1: Factor 45

The largest perfect square factor of 45 is 9 (since \(45 = 9\times5\)).

Step 2: Apply the square root property

\(\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\cdot\sqrt{5}=3\sqrt{5}\)

4. Simplify \(\boldsymbol{\sqrt{54}}\)

Step 1: Factor 54

The largest perfect square factor of 54 is 9 (since \(54 = 9\times6\)).

Step 2: Apply the square root property

\(\sqrt{54}=\sqrt{9\times6}=\sqrt{9}\cdot\sqrt{6}=3\sqrt{6}\)

5. Simplify \(\boldsymbol{\sqrt{56}}\)

Step 1: Factor 56

The largest perfect square factor of 56 is 4 (since \(56 = 4\times14\)).

Step 2: Apply the square root property

\(\sqrt{56}=\sqrt{4\times14}=\sqrt{4}\cdot\sqrt{14}=2\sqrt{14}\)

6. Simplify \(\boldsymbol{\sqrt{60}}\)

Step 1: Factor 60

The largest perfect square factor of 60 is 4 (since \(60 = 4\times15\)).

Step 2: Apply the square root property

\(\sqrt{60}=\sqrt{4\times15}=\sqrt{4}\cdot\sqrt{15}=2\sqrt{15}\)

7. Simplify \(\boldsymbol{\sqrt{75}}\)

Step 1: Factor 75

The largest perfect square factor of 75 is 25 (since \(75 = 25\times3\)).

Step 2: Apply the square root property

\(\sqrt{75}=\sqrt{25\times3}=\sqrt{25}\cdot\sqrt{3}=5\sqrt{3}\)

8. Simplify \(\boldsymbol{\sqrt{80}}\)

Step 1: Factor 80

The largest perfect square factor of 80 is 16 (since \(80 = 16\times5\)).

Step 2: Apply the square root property

\(\sqrt{80}=\sqrt{16\times5}=\sqrt{16}\cdot\sqrt{5}=4\sqrt{5}\)

9. Simplify \(\boldsymbol{\sqrt{90}}\)

Step 1: Factor 90

The largest perfect square factor of 90 is 9 (since \(90 = 9\times10\)).

Step 2: Apply the square root property

\(\sqrt{90}=\sqrt{9\times10}=\sqrt{9}\cdot\sqrt{10}=3\sqrt{10}\)

10. Simplify \(\boldsymbol{\sqrt{108}}\)

Step 1: Factor 108

The largest perfect square factor of 108 is 36 (since \(108 = 36\times3\)).

Step 2: Apply the square root property

\(\sqrt{108}=\sqrt{36\times3}=\sqrt{36}\cdot\sqrt{3}=6\sqrt{3}\)

11. Simplify \(\boldsymbol{\sqrt{125}}\)

Step 1: Factor 125

The largest perfect square factor of 125 is 25 (since \(125 = 25\times5\)).

Step 2: Apply the square root property

\(\sqrt{125}=\sqrt{25\times5}=\sqrt{25}\cdot\sqrt{5}=5\sqrt{5}\)

12. Simplify \(\boldsymbol{\sqrt{150}}\)

Step 1: Factor 150

The largest perfect square factor of 150 is 25 (since \(150 = 25\times6\)).

Step 2: Apply the square root property

\(\sqrt{150}=\sqrt{25\times6}=\sqrt{25}\cdot\sqrt{6}=5\sqrt{6}\)

13. Simplify \(\boldsymbol{\sqrt{x^{3}}}\) (assuming \(x\geq0\))

Step 1: Rewrite \(x^{3}\)

We can rewrite \(x^{3}\) as \(x^{2}\cdot x\) (since \(x^{3}=x^{2}\times x\)).

Step 2: Apply the square root property

\(\sqrt{x^{3}}=\sqrt{x^{2}\cdot x}=\sqrt{x^{2}}\cdot\sqrt{x}\)
Since \(x\geq0\), \(\sqrt{x^{2}} = x\), so \(\sqrt{x^{3}}=x\sqrt{x}\).

14. Simplify \(\boldsymbol{\sqrt{x^{7}}}\) (assuming \(x\geq0\))

Step 1: Rewrite \(x^{7}\)

We can write \(x^{7}\) as \(x^{6}\cdot x\) (because \(x^{7}=x^{6}\times x\) and \(x^{6}=(x^{3})^{2}\) is a perfect square).

Step 2: Apply the square root property

\(\sqrt{x^{7}}=\sqrt{x^{6}\cdot x}=\sqrt{x^{6}}\cdot\sqrt{x}\)
Since \(x\geq0\), \(\sqrt{x^{6}} = x^{3}\), so \(\sqrt{x^{7}}=x^{3}\sqrt{x}\).

15. Simplify \(\boldsymbol{\sqrt{x^{9}}}\) (assuming \(x\geq0\))

Step 1: Rewrite \(x^{9}\)

We can write \(x^{9}\) as \(x^{8}\cdot x\) (because \(x^{9}=x^{8}\times x\) and \(x^{8}=(x^{4})^{2}\) is a perfect square).

Step 2: Apply the square root property

\(\sqrt{x^{9}}=\sqrt{x^{8}\cdot x}=\sqrt{x^{8}}\cdot\sqrt{x}\)
Since \(x\geq0\), \(\sqrt{x^{8}} = x^{4}\), so \(\sqrt{x^{9}}=x^{4}\sqrt{x}\).

Final Answers (for each square root):

• \(\sqrt{40}=\boldsymbol{2\sqrt{10}}\)
• \(\sqrt{44}=\boldsymbol{2\sqrt{11}}\)
• \(\sqrt{45}=\boldsymbol{3\sqrt{5}}\)
• \(\sqrt{54}=\boldsymbol{3\sqrt{6}}\)
• \(\sqrt{56}=\boldsymbol{2\sqrt{14}}\)
• \(\sqrt{60}=\boldsymbol{2\sqrt{15}}\)
• \(\sqrt{75}=\boldsymbol{5\sqrt{3}}\)
• \(\sqrt{80}=\boldsymbol{4\sqrt{5}}\)
• \(\sqrt{90}=\boldsymbol{3\sqrt{10}}\)
• \(\sqrt{108}=\boldsymbol{6\sqrt{3}}\)
• \(\sqrt{125}=\boldsymbol{5\sqrt{5}}\)
• \(\sqrt{150}=\boldsymbol{5\sqrt{6}}\)
• \(\sqrt{x^{3}}=\boldsymbol{x\sqrt{x}}\) (for \(x\geq0\))
• \(\sqrt{x^{7}}=\boldsymbol{x^{3}\sqrt{x}}\) (for \(x\geq0\))
• \(\sqrt{x^{9}}=\boldsymbol{x^{4}\sqrt{x}}\) (for \(x\geq0\))