Question

kuta software - infinite geometry
simplifying square roots
simplify.

1. \\(\sqrt{96}\\)
2. \\(\sqrt{216}\\)
3. \\(\sqrt{98}\\)
4. \\(\sqrt{18}\\)
5. \\(\sqrt{72}\\)
6. \\(\sqrt{144}\\)
7. \\(\sqrt{45}\\)
8. \\(\sqrt{175}\\)
9. \\(\sqrt{343}\\)
10. \\(\sqrt{12}\\)
11. \\(10\sqrt{96}\\)
12. \\(9\sqrt{245}\\)

Explanation:

Let's solve each problem one by one using the property of square roots: \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) where \(a\) and \(b\) are non - negative real numbers, and we try to factor the number inside the square root into a perfect square and another number.

Problem 1: Simplify \(\sqrt{96}\)

Step 1: Factor 96

We know that \(96 = 16\times6\), and \(16\) is a perfect square (\(4^2 = 16\)).

Step 2: Apply the square root property

\(\sqrt{96}=\sqrt{16\times6}=\sqrt{16}\times\sqrt{6}=4\sqrt{6}\)

Problem 2: Simplify \(\sqrt{216}\)

Step 1: Factor 216

\(216 = 36\times6\), and \(36=6^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{216}=\sqrt{36\times6}=\sqrt{36}\times\sqrt{6}=6\sqrt{6}\)

Problem 3: Simplify \(\sqrt{98}\)

Step 1: Factor 98

\(98 = 49\times2\), and \(49 = 7^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{98}=\sqrt{49\times2}=\sqrt{49}\times\sqrt{2}=7\sqrt{2}\)

Problem 4: Simplify \(\sqrt{18}\)

Step 1: Factor 18

\(18 = 9\times2\), and \(9 = 3^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}\)

Problem 5: Simplify \(\sqrt{72}\)

Step 1: Factor 72

\(72=36\times2\), and \(36 = 6^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2}\)

Problem 6: Simplify \(\sqrt{144}\)

We know that \(12^2=144\), so \(\sqrt{144}=12\)

Problem 7: Simplify \(\sqrt{45}\)

Step 1: Factor 45

\(45 = 9\times5\), and \(9 = 3^2\) is a perfect square.

Step 2: Apply the square root property

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

Problem 8: Simplify \(\sqrt{175}\)

Step 1: Factor 175

\(175 = 25\times7\), and \(25 = 5^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{175}=\sqrt{25\times7}=\sqrt{25}\times\sqrt{7}=5\sqrt{7}\)

Problem 9: Simplify \(\sqrt{343}\)

Step 1: Factor 343

\(343=49\times7\), and \(49 = 7^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{343}=\sqrt{49\times7}=\sqrt{49}\times\sqrt{7}=7\sqrt{7}\)

Problem 10: Simplify \(\sqrt{12}\)

Step 1: Factor 12

\(12 = 4\times3\), and \(4 = 2^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}\)

Problem 11: Simplify \(10\sqrt{96}\)

Step 1: Simplify \(\sqrt{96}\) first

From problem 1, we know that \(\sqrt{96}=4\sqrt{6}\)

Step 2: Multiply by 10

\(10\sqrt{96}=10\times4\sqrt{6}=40\sqrt{6}\)

Problem 12: Simplify \(9\sqrt{245}\)

Step 1: Factor 245

\(245 = 49\times5\), and \(49 = 7^2\) is a perfect square. So \(\sqrt{245}=\sqrt{49\times5}=\sqrt{49}\times\sqrt{5}=7\sqrt{5}\)

Step 2: Multiply by 9

\(9\sqrt{245}=9\times7\sqrt{5}=63\sqrt{5}\)

Final Answers:

1. \(\boldsymbol{4\sqrt{6}}\)
2. \(\boldsymbol{6\sqrt{6}}\)
3. \(\boldsymbol{7\sqrt{2}}\)
4. \(\boldsymbol{3\sqrt{2}}\)
5. \(\boldsymbol{6\sqrt{2}}\)
6. \(\boldsymbol{12}\)
7. \(\boldsymbol{3\sqrt{5}}\)
8. \(\boldsymbol{5\sqrt{7}}\)
9. \(\boldsymbol{7\sqrt{7}}\)
10. \(\boldsymbol{2\sqrt{3}}\)
11. \(\boldsymbol{40\sqrt{6}}\)
12. \(\boldsymbol{63\sqrt{5}}\)

Answer:

Let's solve each problem one by one using the property of square roots: \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) where \(a\) and \(b\) are non - negative real numbers, and we try to factor the number inside the square root into a perfect square and another number.

Problem 1: Simplify \(\sqrt{96}\)

Step 1: Factor 96

We know that \(96 = 16\times6\), and \(16\) is a perfect square (\(4^2 = 16\)).

Step 2: Apply the square root property

\(\sqrt{96}=\sqrt{16\times6}=\sqrt{16}\times\sqrt{6}=4\sqrt{6}\)

Problem 2: Simplify \(\sqrt{216}\)

Step 1: Factor 216

\(216 = 36\times6\), and \(36=6^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{216}=\sqrt{36\times6}=\sqrt{36}\times\sqrt{6}=6\sqrt{6}\)

Problem 3: Simplify \(\sqrt{98}\)

Step 1: Factor 98

\(98 = 49\times2\), and \(49 = 7^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{98}=\sqrt{49\times2}=\sqrt{49}\times\sqrt{2}=7\sqrt{2}\)

Problem 4: Simplify \(\sqrt{18}\)

Step 1: Factor 18

\(18 = 9\times2\), and \(9 = 3^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}\)

Problem 5: Simplify \(\sqrt{72}\)

Step 1: Factor 72

\(72=36\times2\), and \(36 = 6^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2}\)

Problem 6: Simplify \(\sqrt{144}\)

We know that \(12^2=144\), so \(\sqrt{144}=12\)

Problem 7: Simplify \(\sqrt{45}\)

Step 1: Factor 45

\(45 = 9\times5\), and \(9 = 3^2\) is a perfect square.

Step 2: Apply the square root property

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

Problem 8: Simplify \(\sqrt{175}\)

Step 1: Factor 175

\(175 = 25\times7\), and \(25 = 5^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{175}=\sqrt{25\times7}=\sqrt{25}\times\sqrt{7}=5\sqrt{7}\)

Problem 9: Simplify \(\sqrt{343}\)

Step 1: Factor 343

\(343=49\times7\), and \(49 = 7^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{343}=\sqrt{49\times7}=\sqrt{49}\times\sqrt{7}=7\sqrt{7}\)

Problem 10: Simplify \(\sqrt{12}\)

Step 1: Factor 12

\(12 = 4\times3\), and \(4 = 2^2\) is a perfect square.

Step 2: Apply the square root property

\(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}\)

Problem 11: Simplify \(10\sqrt{96}\)

Step 1: Simplify \(\sqrt{96}\) first

From problem 1, we know that \(\sqrt{96}=4\sqrt{6}\)

Step 2: Multiply by 10

\(10\sqrt{96}=10\times4\sqrt{6}=40\sqrt{6}\)

Problem 12: Simplify \(9\sqrt{245}\)

Step 1: Factor 245

\(245 = 49\times5\), and \(49 = 7^2\) is a perfect square. So \(\sqrt{245}=\sqrt{49\times5}=\sqrt{49}\times\sqrt{5}=7\sqrt{5}\)

Step 2: Multiply by 9

\(9\sqrt{245}=9\times7\sqrt{5}=63\sqrt{5}\)

Final Answers:

1. \(\boldsymbol{4\sqrt{6}}\)
2. \(\boldsymbol{6\sqrt{6}}\)
3. \(\boldsymbol{7\sqrt{2}}\)
4. \(\boldsymbol{3\sqrt{2}}\)
5. \(\boldsymbol{6\sqrt{2}}\)
6. \(\boldsymbol{12}\)
7. \(\boldsymbol{3\sqrt{5}}\)
8. \(\boldsymbol{5\sqrt{7}}\)
9. \(\boldsymbol{7\sqrt{7}}\)
10. \(\boldsymbol{2\sqrt{3}}\)
11. \(\boldsymbol{40\sqrt{6}}\)
12. \(\boldsymbol{63\sqrt{5}}\)