Simplifying Square Roots worksheet for math practice.

To simplify these square roots, we need to find the largest perfect square that divides into the number inside the square root symbol.

Let's break down the process for each row:

Row 1:
* $\sqrt{18}$: The factors of 18 are 1, 2, 3, 6, 9, 18. The largest perfect square is 9.
* $\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$
* $\sqrt{32}$: The factors of 32 include 16, which is a perfect square.
* $\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$
* $\sqrt{75}$: The factors of 75 include 25, which is a perfect square.
* $\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$
* $\sqrt{24}$: The factors of 24 include 4, which is a perfect square.
* $\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}$

Row 2:
* $\sqrt{45}$: The factors of 45 include 9, which is a perfect square.
* $\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}$
* $\sqrt{80}$: The factors of 80 include 16, which is a perfect square.
* $\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}$
* $\sqrt{28}$: The factors of 28 include 4, which is a perfect square.
* $\sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}$
* $\sqrt{54}$: The factors of 54 include 9, which is a perfect square.
* $\sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6}$

Row 3:
* $\sqrt{96}$: The factors of 96 include 16, which is a perfect square.
* $\sqrt{96} = \sqrt{16 \cdot 6} = \sqrt{16} \cdot \sqrt{6} = 4\sqrt{6}$
* $\sqrt{63}$: The factors of 63 include 9, which is a perfect square.
* $\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}$
* $\sqrt{108}$: The factors of 108 include 36, which is a perfect square.
* $\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}$
* $\sqrt{150}$: The factors of 150 include 25, which is a perfect square.
* $\sqrt{150} = \sqrt{25 \cdot 6} = \sqrt{25} \cdot \sqrt{6} = 5\sqrt{6}$

Row 4: (Note: These already have a number outside the root, we just simplify the part inside)
* $3\sqrt{32}$: Simplify $\sqrt{32}$ first. As found earlier, $\sqrt{32} = 4\sqrt{2}$.
* $3 \cdot 4\sqrt{2} = 12\sqrt{2}$
* $6\sqrt{45}$: Simplify $\sqrt{45}$ first. As found earlier, $\sqrt{45} = 3\sqrt{5}$.
* $6 \cdot 3\sqrt{5} = 18\sqrt{5}$
* $4\sqrt{48}$: Simplify $\sqrt{48}$. The largest perfect square factor is 16. $\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}$.
* $4 \cdot 4\sqrt{3} = 16\sqrt{3}$
* $7\sqrt{108}$: Simplify $\sqrt{108}$ first. As found earlier, $\sqrt{108} = 6\sqrt{3}$.
* $7 \cdot 6\sqrt{3} = 42\sqrt{3}$

Row 5:
* $7\sqrt{64}$: $\sqrt{64}$ is a perfect square, it equals 8.
* $7 \cdot 8 = 56$
* $8\sqrt{20}$: Simplify $\sqrt{20}$. The largest perfect square factor is 4. $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$.
* $8 \cdot 2\sqrt{5} = 16\sqrt{5}$
* $3\sqrt{144}$: $\sqrt{144}$ is a perfect square, it equals 12.
* $3 \cdot 12 = 36$
* $6\sqrt{72}$: Simplify $\sqrt{72}$. The largest perfect square factor is 36. $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.
* $6 \cdot 6\sqrt{2} = 36\sqrt{2}$

Final Answer:
Row 1: $3\sqrt{2}$, $4\sqrt{2}$, $5\sqrt{3}$, $2\sqrt{6}$
Row 2: $3\sqrt{5}$, $4\sqrt{5}$, $2\sqrt{7}$, $3\sqrt{6}$
Row 3: $4\sqrt{6}$, $3\sqrt{7}$, $6\sqrt{3}$, $5\sqrt{6}$
Row 4: $12\sqrt{2}$, $18\sqrt{5}$, $16\sqrt{3}$, $42\sqrt{3}$
Row 5: $56$, $16\sqrt{5}$, $36$, $36\sqrt{2}$