Question

$\sqrt{28}$ $\sqrt{32}$ $\sqrt{50}$ $\sqrt{52}$ $\sqrt{68}$ $\sqrt{72}$ $\sqrt{98}$ $\sqrt{99}$ $\sqrt{200}$ $\sqrt{x^3}$

Answer:

To simplify these square roots, we use the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a\geq0,b\geq0\)) to factor out perfect squares from the radicand. Let's simplify each one:

1. Simplify \(\boldsymbol{\sqrt{28}}\)

Step1: Factor 28 into perfect square and other factor

28 can be factored as \(4\times7\), where 4 is a perfect square (\(4 = 2^2\)). So we have:
\(\sqrt{28}=\sqrt{4\times7}\)

Step2: Apply the square - root property

Using \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), we get:
\(\sqrt{4\times7}=\sqrt{4}\times\sqrt{7}=2\sqrt{7}\)

2. Simplify \(\boldsymbol{\sqrt{32}}\)

Step1: Factor 32 into perfect square and other factor

32 can be factored as \(16\times2\), where 16 is a perfect square (\(16 = 4^2\)). So:
\(\sqrt{32}=\sqrt{16\times2}\)

Step2: Apply the square - root property

\(\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}=4\sqrt{2}\)

3. Simplify \(\boldsymbol{\sqrt{50}}\)

Step1: Factor 50 into perfect square and other factor

50 can be factored as \(25\times2\), where 25 is a perfect square (\(25 = 5^2\)). So:
\(\sqrt{50}=\sqrt{25\times2}\)

Step2: Apply the square - root property

\(\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\)

4. Simplify \(\boldsymbol{\sqrt{52}}\)

Step1: Factor 52 into perfect square and other factor

52 can be factored as \(4\times13\), where 4 is a perfect square (\(4 = 2^2\)). So:
\(\sqrt{52}=\sqrt{4\times13}\)

Step2: Apply the square - root property

\(\sqrt{4\times13}=\sqrt{4}\times\sqrt{13}=2\sqrt{13}\)

5. Simplify \(\boldsymbol{\sqrt{68}}\)

Step1: Factor 68 into perfect square and other factor

68 can be factored as \(4\times17\), where 4 is a perfect square (\(4 = 2^2\)). So:
\(\sqrt{68}=\sqrt{4\times17}\)

Step2: Apply the square - root property

\(\sqrt{4\times17}=\sqrt{4}\times\sqrt{17}=2\sqrt{17}\)

6. Simplify \(\boldsymbol{\sqrt{72}}\)

Step1: Factor 72 into perfect square and other factor

72 can be factored as \(36\times2\), where 36 is a perfect square (\(36 = 6^2\)). So:
\(\sqrt{72}=\sqrt{36\times2}\)

Step2: Apply the square - root property

\(\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2}\)

7. Simplify \(\boldsymbol{\sqrt{98}}\)

Step1: Factor 98 into perfect square and other factor

98 can be factored as \(49\times2\), where 49 is a perfect square (\(49 = 7^2\)). So:
\(\sqrt{98}=\sqrt{49\times2}\)

Step2: Apply the square - root property

\(\sqrt{49\times2}=\sqrt{49}\times\sqrt{2}=7\sqrt{2}\)

8. Simplify \(\boldsymbol{\sqrt{99}}\)

Step1: Factor 99 into perfect square and other factor

99 can be factored as \(9\times11\), where 9 is a perfect square (\(9 = 3^2\)). So:
\(\sqrt{99}=\sqrt{9\times11}\)

Step2: Apply the square - root property

\(\sqrt{9\times11}=\sqrt{9}\times\sqrt{11}=3\sqrt{11}\)

9. Simplify \(\boldsymbol{\sqrt{200}}\)

Step1: Factor 200 into perfect square and other factor

200 can be factored as \(100\times2\), where 100 is a perfect square (\(100 = 10^2\)). So:
\(\sqrt{200}=\sqrt{100\times2}\)

Step2: Apply the square - root property

\(\sqrt{100\times2}=\sqrt{100}\times\sqrt{2}=10\sqrt{2}\)

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

Step1: Rewrite \(x^{3}\) as a product

We can rewrite \(x^{3}\) as \(x^{2}\times x\), where \(x^{2}\) is a perfect square (since for \(x\geq0\), \((\sqrt{x^{2}})=x\)). So:
\(\sqrt{x^{3}}=\sqrt{x^{2}\times x}\)

Step2: Apply the square - root property

\(\sqrt{x^{2}\times x}=\sqrt{x^{2}}\times\sqrt{x}=x\sqrt{x}\)

Final Answers:

\(\sqrt{28}=\boldsymbol{2\sqrt{7}}\); \(\sqrt{32}=\boldsymbol{4\sqrt{2}}\); \(\sqrt{50}=\boldsymbol{5\sqrt{2}}\); \(\sqrt{52}=\boldsymbol{2\sqrt{13}}\); \(\sqrt{68}=\boldsymbol{2\sqrt{17}}\); \(\sqrt{72}=\boldsymbol{6\sqrt{2}}\); \(\sqrt{98}=\boldsymbol{7\sqrt{2}}\); \(\sqrt{99}=\boldsymbol{3\sqrt{11}}\); \(\sqrt{200}=\boldsymbol{10\sqrt{2}}\); \(\sqrt{x^{3}}=\boldsymbol{x\sqrt{x}}\) (for \(x\geq0\))