Simplifying Radical Expressions worksheet with 14 problems for practice.

To simplify these radical expressions, we need to break down the numbers and variables inside the square roots into perfect squares. A perfect square is a number that results from multiplying an integer by itself (like 4, 9, 16, 25, etc.) or a variable with an even exponent (like $x^2$, $y^4$).

Here is the step-by-step breakdown for each problem:

1) $\sqrt{125n}$
* Break 125 into factors where one is a perfect square: $125 = 25 \times 5$.
* $\sqrt{25 \times 5 \times n} = \sqrt{25} \times \sqrt{5n}$
* $\sqrt{25} = 5$
* Answer: $5\sqrt{5n}$

2) $\sqrt{216v}$
* Find the largest perfect square factor of 216. $216 = 36 \times 6$.
* $\sqrt{36 \times 6 \times v} = \sqrt{36} \times \sqrt{6v}$
* $\sqrt{36} = 6$
* Answer: $6\sqrt{6v}$

3) $\sqrt{512k^2}$
* Find the largest perfect square factor of 512. $512 = 256 \times 2$.
* $\sqrt{256 \times 2 \times k^2} = \sqrt{256} \times \sqrt{2} \times \sqrt{k^2}$
* $\sqrt{256} = 16$ and $\sqrt{k^2} = k$.
* Answer: $16k\sqrt{2}$

4) $\sqrt{512m^3}$
* Break $m^3$ into $m^2 \times m$.
* Use the number factor from problem 3: $512 = 256 \times 2$.
* $\sqrt{256 \times 2 \times m^2 \times m} = \sqrt{256} \times \sqrt{m^2} \times \sqrt{2m}$
* $16 \times m \times \sqrt{2m}$
* Answer: $16m\sqrt{2m}$

5) $\sqrt{216k^4}$
* From problem 2, we know $216 = 36 \times 6$.
* $\sqrt{36 \times 6 \times k^4} = \sqrt{36} \times \sqrt{6} \times \sqrt{k^4}$
* $\sqrt{36} = 6$ and $\sqrt{k^4} = k^2$.
* Answer: $6k^2\sqrt{6}$

6) $\sqrt{100v^3}$
* $100$ is a perfect square ($10^2$).
* Break $v^3$ into $v^2 \times v$.
* $\sqrt{100 \times v^2 \times v} = \sqrt{100} \times \sqrt{v^2} \times \sqrt{v}$
* $10 \times v \times \sqrt{v}$
* Answer: $10v\sqrt{v}$

7) $\sqrt{80p^3}$
* Find the largest perfect square factor of 80. $80 = 16 \times 5$.
* Break $p^3$ into $p^2 \times p$.
* $\sqrt{16 \times 5 \times p^2 \times p} = \sqrt{16} \times \sqrt{p^2} \times \sqrt{5p}$
* $4 \times p \times \sqrt{5p}$
* Answer: $4p\sqrt{5p}$

8) $\sqrt{45p^2}$
* Find the largest perfect square factor of 45. $45 = 9 \times 5$.
* $\sqrt{9 \times 5 \times p^2} = \sqrt{9} \times \sqrt{p^2} \times \sqrt{5}$
* $3 \times p \times \sqrt{5}$
* Answer: $3p\sqrt{5}$

9) $\sqrt{147m^3}$
* Find the largest perfect square factor of 147. $147 = 49 \times 3$.
* Break $m^3$ into $m^2 \times m$.
* $\sqrt{49 \times 3 \times m^2 \times m} = \sqrt{49} \times \sqrt{m^2} \times \sqrt{3m}$
* $7 \times m \times \sqrt{3m}$
* Answer: $7m\sqrt{3m}$

10) $\sqrt{100m^4}$
* $100$ is a perfect square ($10^2$).
* $\sqrt{m^4}$ is $m^2$ because $(m^2)^2 = m^4$.
* $10 \times m^2$
* Answer: $10m^2$

11) $\sqrt{75x^2}$
* Find the largest perfect square factor of 75. $75 = 25 \times 3$.
* $\sqrt{25 \times 3 \times x^2} = \sqrt{25} \times \sqrt{x^2} \times \sqrt{3}$
* $5 \times x \times \sqrt{3}$
* Answer: $5x\sqrt{3}$

12) $\sqrt{64m^4}$
* $64$ is a perfect square ($8^2$).
* $\sqrt{m^4}$ is $m^2$.
* $8 \times m^2$
* Answer: $8m^2$

13) $\sqrt{16u^4}$
* $16$ is a perfect square ($4^2$).
* $\sqrt{u^4}$ is $u^2$.
* $4 \times u^2$
* Answer: $4u^2$

14) $\sqrt{28x^3}$
* Find the largest perfect square factor of 28. $28 = 4 \times 7$.
* Break $x^3$ into $x^2 \times x$.
* $\sqrt{4 \times 7 \times x^2 \times x} = \sqrt{4} \times \sqrt{x^2} \times \sqrt{7x}$
* $2 \times x \times \sqrt{7x}$
* Answer: $2x\sqrt{7x}$

Final Answer:
1) $5\sqrt{5n}$
2) $6\sqrt{6v}$
3) $16k\sqrt{2}$
4) $16m\sqrt{2m}$
5) $6k^2\sqrt{6}$
6) $10v\sqrt{v}$
7) $4p\sqrt{5p}$
8) $3p\sqrt{5}$
9) $7m\sqrt{3m}$
10) $10m^2$
11) $5x\sqrt{3}$
12) $8m^2$
13) $4u^2$
14) $2x\sqrt{7x}$