Simplifying radical expressions :: Algebra Helper - Free Printable
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Step-by-step solution for: Simplifying radical expressions :: Algebra Helper
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Show Answer Key & Explanations
Step-by-step solution for: Simplifying radical expressions :: Algebra Helper
Here are the simplified forms for each radical expression. To solve these, we look for perfect square factors inside the square root and pull them out.
1. $\sqrt{18}$
$\sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$
2. $\sqrt{125}$
$\sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}$
3. $\sqrt{72}$
$\sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$
4. $\sqrt{180}$
$\sqrt{36 \cdot 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5}$
5. $\sqrt{a^3}$
$\sqrt{a^2 \cdot a} = \sqrt{a^2} \cdot \sqrt{a} = a\sqrt{a}$
6. $\sqrt{b^7}$
$\sqrt{b^6 \cdot b} = \sqrt{b^6} \cdot \sqrt{b} = b^3\sqrt{b}$
*(Note: $b^6$ is a perfect square because the exponent 6 is even. Half of 6 is 3.)*
7. $\sqrt{m^{11}}$
$\sqrt{m^{10} \cdot m} = \sqrt{m^{10}} \cdot \sqrt{m} = m^5\sqrt{m}$
8. $\sqrt{75x^7y^5}$
Break it down: $\sqrt{25 \cdot 3 \cdot x^6 \cdot x \cdot y^4 \cdot y}$
Pull out perfect squares: $\sqrt{25} \cdot \sqrt{x^6} \cdot \sqrt{y^4} \cdot \sqrt{3xy}$
Result: $5x^3y^2\sqrt{3xy}$
9. $\sqrt{27a^{11}b^7}$
Break it down: $\sqrt{9 \cdot 3 \cdot a^{10} \cdot a \cdot b^6 \cdot b}$
Pull out perfect squares: $\sqrt{9} \cdot \sqrt{a^{10}} \cdot \sqrt{b^6} \cdot \sqrt{3ab}$
Result: $3a^5b^3\sqrt{3ab}$
10. $\sqrt{32a^7b^4}$
Break it down: $\sqrt{16 \cdot 2 \cdot a^6 \cdot a \cdot b^4}$
Pull out perfect squares: $\sqrt{16} \cdot \sqrt{a^6} \cdot \sqrt{b^4} \cdot \sqrt{2a}$
Result: $4a^3b^2\sqrt{2a}$
11. $\sqrt{9a^8}$
$\sqrt{9} \cdot \sqrt{a^8} = 3a^4$
12. $\sqrt{45a^7}$
Break it down: $\sqrt{9 \cdot 5 \cdot a^6 \cdot a}$
Pull out perfect squares: $\sqrt{9} \cdot \sqrt{a^6} \cdot \sqrt{5a}$
Result: $3a^3\sqrt{5a}$
13. $\sqrt{36x^2y^6}$
$\sqrt{36} \cdot \sqrt{x^2} \cdot \sqrt{y^6} = 6xy^3$
14. $\sqrt{12x^{20}y^8}$
Break it down: $\sqrt{4 \cdot 3 \cdot x^{20} \cdot y^8}$
Pull out perfect squares: $\sqrt{4} \cdot \sqrt{x^{20}} \cdot \sqrt{y^8} \cdot \sqrt{3}$
Result: $2x^{10}y^4\sqrt{3}$
15. $-\sqrt{200}$
Keep the negative sign outside. Simplify $\sqrt{200}$:
$\sqrt{100 \cdot 2} = 10\sqrt{2}$
Result: $-10\sqrt{2}$
16. $\sqrt{196}$
$14 \cdot 14 = 196$, so this is a perfect square.
Result: $14$
17. $\sqrt{63x^4y}$
Break it down: $\sqrt{9 \cdot 7 \cdot x^4 \cdot y}$
Pull out perfect squares: $\sqrt{9} \cdot \sqrt{x^4} \cdot \sqrt{7y}$
Result: $3x^2\sqrt{7y}$
18. $\sqrt{6x^3}$
Break it down: $\sqrt{6 \cdot x^2 \cdot x}$
Pull out perfect squares: $\sqrt{x^2} \cdot \sqrt{6x}$
Result: $x\sqrt{6x}$
19. $\sqrt{100x^5y}$
Break it down: $\sqrt{100 \cdot x^4 \cdot x \cdot y}$
Pull out perfect squares: $\sqrt{100} \cdot \sqrt{x^4} \cdot \sqrt{xy}$
Result: $10x^2\sqrt{xy}$
20. $\sqrt{80x^{100}y^{49}}$
Break it down: $\sqrt{16 \cdot 5 \cdot x^{100} \cdot y^{48} \cdot y}$
Pull out perfect squares: $\sqrt{16} \cdot \sqrt{x^{100}} \cdot \sqrt{y^{48}} \cdot \sqrt{5y}$
Result: $4x^{50}y^{24}\sqrt{5y}$
Final Answer:
1. $3\sqrt{2}$
2. $5\sqrt{5}$
3. $6\sqrt{2}$
4. $6\sqrt{5}$
5. $a\sqrt{a}$
6. $b^3\sqrt{b}$
7. $m^5\sqrt{m}$
8. $5x^3y^2\sqrt{3xy}$
9. $3a^5b^3\sqrt{3ab}$
10. $4a^3b^2\sqrt{2a}$
11. $3a^4$
12. $3a^3\sqrt{5a}$
13. $6xy^3$
14. $2x^{10}y^4\sqrt{3}$
15. $-10\sqrt{2}$
16. $14$
17. $3x^2\sqrt{7y}$
18. $x\sqrt{6x}$
19. $10x^2\sqrt{xy}$
20. $4x^{50}y^{24}\sqrt{5y}$
1. $\sqrt{18}$
$\sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$
2. $\sqrt{125}$
$\sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}$
3. $\sqrt{72}$
$\sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$
4. $\sqrt{180}$
$\sqrt{36 \cdot 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5}$
5. $\sqrt{a^3}$
$\sqrt{a^2 \cdot a} = \sqrt{a^2} \cdot \sqrt{a} = a\sqrt{a}$
6. $\sqrt{b^7}$
$\sqrt{b^6 \cdot b} = \sqrt{b^6} \cdot \sqrt{b} = b^3\sqrt{b}$
*(Note: $b^6$ is a perfect square because the exponent 6 is even. Half of 6 is 3.)*
7. $\sqrt{m^{11}}$
$\sqrt{m^{10} \cdot m} = \sqrt{m^{10}} \cdot \sqrt{m} = m^5\sqrt{m}$
8. $\sqrt{75x^7y^5}$
Break it down: $\sqrt{25 \cdot 3 \cdot x^6 \cdot x \cdot y^4 \cdot y}$
Pull out perfect squares: $\sqrt{25} \cdot \sqrt{x^6} \cdot \sqrt{y^4} \cdot \sqrt{3xy}$
Result: $5x^3y^2\sqrt{3xy}$
9. $\sqrt{27a^{11}b^7}$
Break it down: $\sqrt{9 \cdot 3 \cdot a^{10} \cdot a \cdot b^6 \cdot b}$
Pull out perfect squares: $\sqrt{9} \cdot \sqrt{a^{10}} \cdot \sqrt{b^6} \cdot \sqrt{3ab}$
Result: $3a^5b^3\sqrt{3ab}$
10. $\sqrt{32a^7b^4}$
Break it down: $\sqrt{16 \cdot 2 \cdot a^6 \cdot a \cdot b^4}$
Pull out perfect squares: $\sqrt{16} \cdot \sqrt{a^6} \cdot \sqrt{b^4} \cdot \sqrt{2a}$
Result: $4a^3b^2\sqrt{2a}$
11. $\sqrt{9a^8}$
$\sqrt{9} \cdot \sqrt{a^8} = 3a^4$
12. $\sqrt{45a^7}$
Break it down: $\sqrt{9 \cdot 5 \cdot a^6 \cdot a}$
Pull out perfect squares: $\sqrt{9} \cdot \sqrt{a^6} \cdot \sqrt{5a}$
Result: $3a^3\sqrt{5a}$
13. $\sqrt{36x^2y^6}$
$\sqrt{36} \cdot \sqrt{x^2} \cdot \sqrt{y^6} = 6xy^3$
14. $\sqrt{12x^{20}y^8}$
Break it down: $\sqrt{4 \cdot 3 \cdot x^{20} \cdot y^8}$
Pull out perfect squares: $\sqrt{4} \cdot \sqrt{x^{20}} \cdot \sqrt{y^8} \cdot \sqrt{3}$
Result: $2x^{10}y^4\sqrt{3}$
15. $-\sqrt{200}$
Keep the negative sign outside. Simplify $\sqrt{200}$:
$\sqrt{100 \cdot 2} = 10\sqrt{2}$
Result: $-10\sqrt{2}$
16. $\sqrt{196}$
$14 \cdot 14 = 196$, so this is a perfect square.
Result: $14$
17. $\sqrt{63x^4y}$
Break it down: $\sqrt{9 \cdot 7 \cdot x^4 \cdot y}$
Pull out perfect squares: $\sqrt{9} \cdot \sqrt{x^4} \cdot \sqrt{7y}$
Result: $3x^2\sqrt{7y}$
18. $\sqrt{6x^3}$
Break it down: $\sqrt{6 \cdot x^2 \cdot x}$
Pull out perfect squares: $\sqrt{x^2} \cdot \sqrt{6x}$
Result: $x\sqrt{6x}$
19. $\sqrt{100x^5y}$
Break it down: $\sqrt{100 \cdot x^4 \cdot x \cdot y}$
Pull out perfect squares: $\sqrt{100} \cdot \sqrt{x^4} \cdot \sqrt{xy}$
Result: $10x^2\sqrt{xy}$
20. $\sqrt{80x^{100}y^{49}}$
Break it down: $\sqrt{16 \cdot 5 \cdot x^{100} \cdot y^{48} \cdot y}$
Pull out perfect squares: $\sqrt{16} \cdot \sqrt{x^{100}} \cdot \sqrt{y^{48}} \cdot \sqrt{5y}$
Result: $4x^{50}y^{24}\sqrt{5y}$
Final Answer:
1. $3\sqrt{2}$
2. $5\sqrt{5}$
3. $6\sqrt{2}$
4. $6\sqrt{5}$
5. $a\sqrt{a}$
6. $b^3\sqrt{b}$
7. $m^5\sqrt{m}$
8. $5x^3y^2\sqrt{3xy}$
9. $3a^5b^3\sqrt{3ab}$
10. $4a^3b^2\sqrt{2a}$
11. $3a^4$
12. $3a^3\sqrt{5a}$
13. $6xy^3$
14. $2x^{10}y^4\sqrt{3}$
15. $-10\sqrt{2}$
16. $14$
17. $3x^2\sqrt{7y}$
18. $x\sqrt{6x}$
19. $10x^2\sqrt{xy}$
20. $4x^{50}y^{24}\sqrt{5y}$
Parent Tip: Review the logic above to help your child master the concept of simplifying square roots with variables worksheet.
