Question

multiplying square roots
date
simplify.

1. \\(sqrt{5} cdot sqrt{5}\\) \t\t\t\t2) \\(sqrt{10} cdot sqrt{2}\\)

1. \\(sqrt{8} cdot sqrt{8}\\) \t\t\t\t4) \\(sqrt{20} cdot sqrt{10}\\)

1. \\(sqrt{3} cdot sqrt{3}\\) \t\t\t\t6) \\(sqrt{5} cdot sqrt{12}\\)

1. \\(2sqrt{2} cdot sqrt{12}\\) \t\t\t\t8) \\(sqrt{5} cdot 2sqrt{2}\\)

\t\t\t\t\t\t\t\t\\(2sqrt{10}\\)

1. \\(sqrt{6} cdot -2sqrt{6}\\) \t\t\t10) \\(sqrt{2} cdot -2sqrt{5}\\)

1. \\(sqrt{6} cdot -sqrt{9}\\) \t\t\t12) \\(sqrt{5} cdot -5sqrt{5}\\)

Answer:

Let's solve each problem step by step using the property of square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ (for $a \geq 0, b \geq 0$) and also handling coefficients.

1) $\boldsymbol{\sqrt{5} \cdot \sqrt{5}}$

Step1: Apply the square root multiplication property

Using $\sqrt{a} \cdot \sqrt{a} = a$ (since $\sqrt{a} \cdot \sqrt{a} = \sqrt{a \cdot a} = \sqrt{a^2} = a$ for $a \geq 0$). Here $a = 5$.
So, $\sqrt{5} \cdot \sqrt{5} = 5$

2) $\boldsymbol{\sqrt{10} \cdot \sqrt{2}}$

Step1: Apply the square root multiplication property

$\sqrt{10} \cdot \sqrt{2} = \sqrt{10 \times 2}$

Step2: Simplify the product inside the square root

$10 \times 2 = 20$, so $\sqrt{20}$

Step3: Simplify $\sqrt{20}$

Factor 20 as $4 \times 5$, and $\sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}$

3) $\boldsymbol{\sqrt{8} \cdot \sqrt{8}}$

Step1: Apply the square root multiplication property (similar to problem 1)

$\sqrt{8} \cdot \sqrt{8} = 8$ (since $\sqrt{a} \cdot \sqrt{a} = a$ for $a \geq 0$)

4) $\boldsymbol{\sqrt{20} \cdot \sqrt{10}}$

Step1: Apply the square root multiplication property

$\sqrt{20} \cdot \sqrt{10} = \sqrt{20 \times 10}$

Step2: Simplify the product inside the square root

$20 \times 10 = 200$

Step3: Simplify $\sqrt{200}$

Factor 200 as $100 \times 2$, so $\sqrt{100 \times 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$

5) $\boldsymbol{\sqrt{3} \cdot \sqrt{3}}$

Step1: Apply the square root multiplication property (similar to problem 1)

$\sqrt{3} \cdot \sqrt{3} = 3$ (since $\sqrt{a} \cdot \sqrt{a} = a$ for $a \geq 0$)

6) $\boldsymbol{\sqrt{5} \cdot \sqrt{12}}$

Step1: Apply the square root multiplication property

$\sqrt{5} \cdot \sqrt{12} = \sqrt{5 \times 12}$

Step2: Simplify the product inside the square root

$5 \times 12 = 60$

Step3: Simplify $\sqrt{60}$

Factor 60 as $4 \times 15$, so $\sqrt{4 \times 15} = \sqrt{4} \cdot \sqrt{15} = 2\sqrt{15}$

7) $\boldsymbol{2\sqrt{2} \cdot \sqrt{12}}$

Step1: Multiply the coefficients and the square roots separately

$2\sqrt{2} \cdot \sqrt{12} = 2 \times (\sqrt{2} \cdot \sqrt{12})$

Step2: Apply the square root multiplication property to the square roots

$\sqrt{2} \cdot \sqrt{12} = \sqrt{2 \times 12} = \sqrt{24}$

Step3: Simplify $\sqrt{24}$

Factor 24 as $4 \times 6$, so $\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}$

Step4: Multiply by the coefficient 2

$2 \times 2\sqrt{6} = 4\sqrt{6}$

8) $\boldsymbol{\sqrt{5} \cdot 2\sqrt{2}}$

Step1: Multiply the coefficients and the square roots separately

$\sqrt{5} \cdot 2\sqrt{2} = 2 \times (\sqrt{5} \cdot \sqrt{2})$

Step2: Apply the square root multiplication property

$\sqrt{5} \cdot \sqrt{2} = \sqrt{5 \times 2} = \sqrt{10}$

Step3: Multiply by the coefficient 2

$2 \times \sqrt{10} = 2\sqrt{10}$ (matches the given note)

9) $\boldsymbol{\sqrt{6} \cdot -2\sqrt{6}}$

Step1: Multiply the coefficients and the square roots separately

$\sqrt{6} \cdot -2\sqrt{6} = -2 \times (\sqrt{6} \cdot \sqrt{6})$

Step2: Apply the square root multiplication property (similar to problem 1)

$\sqrt{6} \cdot \sqrt{6} = 6$

Step3: Multiply by the coefficient -2

$-2 \times 6 = -12$

10) $\boldsymbol{\sqrt{2} \cdot -2\sqrt{5}}$

Step1: Multiply the coefficients and the square roots separately

$\sqrt{2} \cdot -2\sqrt{5} = -2 \times (\sqrt{2} \cdot \sqrt{5})$

Step2: Apply the square root multiplication property

$\sqrt{2} \cdot \sqrt{5} = \sqrt{2 \times 5} = \sqrt{10}$

Step3: Multiply by the coefficient -2

$-2 \times \sqrt{10} = -2\sqrt{10}$

11) $\boldsymbol{\sqrt{6} \cdot -\sqrt{9}}$

Step1: Simplify $\sqrt{9}$

$\sqrt{9} = 3$

Step2: Multiply with $\sqrt{6}$ and the coefficient -1

$\sqrt{6} \cdot -3 = -3\sqrt{6}$

12) $\boldsymbol{\sqrt{5} \cdot -5\sqrt{5}}$

Step1: Multiply the coefficients and the square roots separately

$\sqrt{5} \cdot -5\sqrt{5} = -5 \times (\sqrt{5} \cdot \sqrt{5})$

Step2: Apply the square root multiplication property (similar to problem 1)

$\sqrt{5} \cdot \sqrt{5} = 5$

Step3: Multiply by the coefficient -5

$-5 \times 5 = -25$

Final Answers:

1. $\boldsymbol{5}$
2. $\boldsymbol{2\sqrt{5}}$
3. $\boldsymbol{8}$
4. $\boldsymbol{10\sqrt{2}}$
5. $\boldsymbol{3}$
6. $\boldsymbol{2\sqrt{15}}$
7. $\boldsymbol{4\sqrt{6}}$
8. $\boldsymbol{2\sqrt{10}}$
9. $\boldsymbol{-12}$
10. $\boldsymbol{-2\sqrt{10}}$
11. $\boldsymbol{-3\sqrt{6}}$
12. $\boldsymbol{-25}$