Multiplication and Division Worksheets with Answer Key - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1) $\sqrt{3} \cdot \sqrt{3}$
* Step 1: When you multiply two square roots with the same number inside, they cancel out the square root. Alternatively, think of it as $\sqrt{3 \cdot 3} = \sqrt{9}$.
* Step 2: The square root of 9 is 3.
* Answer: $3$

2) $4\sqrt{15} \cdot \sqrt{3}$
* Step 1: Multiply the numbers inside the radicals: $15 \cdot 3 = 45$. Keep the 4 outside. So, we have $4\sqrt{45}$.
* Step 2: Simplify $\sqrt{45}$. Look for a perfect square factor. $45 = 9 \cdot 5$.
* Step 3: $\sqrt{45} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}$.
* Step 4: Multiply this by the 4 that was already outside: $4 \cdot 3\sqrt{5} = 12\sqrt{5}$.
* Answer: $12\sqrt{5}$

3) $\sqrt{10} \cdot \sqrt{3}$
* Step 1: Multiply the numbers inside: $10 \cdot 3 = 30$.
* Step 2: Check if 30 has any perfect square factors (like 4, 9, 16, 25). It does not ($30 = 2 \cdot 3 \cdot 5$).
* Answer: $\sqrt{30}$

4) $\sqrt{3} \cdot \sqrt{6}$
* Step 1: Multiply the insides: $3 \cdot 6 = 18$. So, $\sqrt{18}$.
* Step 2: Simplify $\sqrt{18}$. $18 = 9 \cdot 2$.
* Step 3: $\sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$.
* Answer: $3\sqrt{2}$

5) $\sqrt{5} \cdot \sqrt{3}$
* Step 1: Multiply the insides: $5 \cdot 3 = 15$.
* Step 2: 15 has no perfect square factors.
* Answer: $\sqrt{15}$

6) $\sqrt{6} \cdot \sqrt{6}$
* Step 1: Just like problem #1, multiplying a square root by itself removes the radical. $\sqrt{36}$.
* Step 2: The square root of 36 is 6.
* Answer: $6$

7) $\frac{\sqrt{36}}{\sqrt{4}}$
* Step 1: You can solve the top and bottom separately because both are perfect squares.
* Step 2: $\sqrt{36} = 6$ and $\sqrt{4} = 2$.
* Step 3: Divide: $\frac{6}{2} = 3$.
* Answer: $3$

8) $\frac{\sqrt{25}}{\sqrt{9}}$
* Step 1: Solve the top and bottom separately.
* Step 2: $\sqrt{25} = 5$ and $\sqrt{9} = 3$.
* Step 3: Put them back into the fraction. It cannot be simplified further.
* Answer: $\frac{5}{3}$

9) $\frac{2\sqrt{20}}{\sqrt{4}}$
* Step 1: Simplify the denominator first. $\sqrt{4} = 2$.
* Step 2: Now the expression is $\frac{2\sqrt{20}}{2}$.
* Step 3: The 2 on top and the 2 on the bottom cancel out, leaving just $\sqrt{20}$.
* Step 4: Simplify $\sqrt{20}$. $20 = 4 \cdot 5$. So, $\sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}$.
* Answer: $2\sqrt{5}$

10) $\frac{-3 - \sqrt{2}}{\sqrt{3}}$
* Step 1: To remove the square root from the bottom (rationalize), multiply the top and bottom by $\sqrt{3}$.
* Step 2: Bottom: $\sqrt{3} \cdot \sqrt{3} = 3$.
* Step 3: Top: Distribute $\sqrt{3}$ to both parts.
* $-3 \cdot \sqrt{3} = -3\sqrt{3}$
* $-\sqrt{2} \cdot \sqrt{3} = -\sqrt{6}$
* Step 4: Combine them over the new denominator.
* Answer: $\frac{-3\sqrt{3} - \sqrt{6}}{3}$

11) $\frac{5}{\sqrt{7}}$
* Step 1: Rationalize the denominator by multiplying top and bottom by $\sqrt{7}$.
* Step 2: Bottom: $\sqrt{7} \cdot \sqrt{7} = 7$.
* Step 3: Top: $5 \cdot \sqrt{7} = 5\sqrt{7}$.
* Answer: $\frac{5\sqrt{7}}{7}$

12) $\frac{4\sqrt{28}}{\sqrt{4}}$
* Step 1: Simplify the denominator: $\sqrt{4} = 2$.
* Step 2: The expression becomes $\frac{4\sqrt{28}}{2}$.
* Step 3: Divide 4 by 2, which leaves $2\sqrt{28}$.
* Step 4: Simplify $\sqrt{28}$. $28 = 4 \cdot 7$. So $\sqrt{28} = 2\sqrt{7}$.
* Step 5: Multiply by the 2 from Step 3: $2 \cdot (2\sqrt{7}) = 4\sqrt{7}$.
* Answer: $4\sqrt{7}$

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Final Answer:
1) $3$
2) $12\sqrt{5}$
3) $\sqrt{30}$
4) $3\sqrt{2}$
5) $\sqrt{15}$
6) $6$
7) $3$
8) $\frac{5}{3}$
9) $2\sqrt{5}$
10) $\frac{-3\sqrt{3} - \sqrt{6}}{3}$
11) $\frac{5\sqrt{7}}{7}$
12) $4\sqrt{7}$