Question

ready topic simplifying radicals a very common radical expression is a square root. one way to think of a square root is the number that will multiply by itself to create a desired value. for example: √2 is the number that will multiply by itself to equal 2. and in like manner √16 is the number that will multiply by itself to equal 16, in this case the value is 4 because 4×4 = 16. (when the square root of a square number is taken you get a nice whole - number value. otherwise an irrational number is produced.) this same pattern holds true for other radicals such as cube roots and fourth roots and so forth. for example ∛8 is the number that will multiply by itself three times to equal 8. in this case it is equal to the value of 2 because 2³=2×2×2 = 8. with this in mind radicals can be simplified. see the examples below. example 1: simplify √20 √20 = √4·5 = √2·2·5 = 2√5 example 2: simplify ∛96 ∛96 = ∛2⁵·3 = 2∛3 simplify each of the radicals. 1. √40 2. √50 3. ∛16 4. √72 5. ∜81 6. √32 7. ∛160 8. √45 9. ∛54

Explanation:

Step1: Factor the number inside radical

For $\sqrt{40}$, factor 40 as $4\times10$. Then $\sqrt{40}=\sqrt{4\times10}=\sqrt{4}\times\sqrt{10}=2\sqrt{10}$.

Step2: Factor the number inside radical

For $\sqrt{50}$, factor 50 as $25\times2$. Then $\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}$.

Step3: Factor the number inside radical

For $\sqrt[3]{16}$, factor 16 as $8\times2$. Then $\sqrt[3]{16}=\sqrt[3]{8\times2}=\sqrt[3]{8}\times\sqrt[3]{2}=2\sqrt[3]{2}$.

Step4: Factor the number inside radical

For $\sqrt{72}$, factor 72 as $36\times2$. Then $\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2}$.

Step5: Factor the number inside radical

For $\sqrt[4]{81}$, since $3^4 = 81$, $\sqrt[4]{81}=3$.

Step6: Factor the number inside radical

For $\sqrt{32}$, factor 32 as $16\times2$. Then $\sqrt{32}=\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}=4\sqrt{2}$.

Step7: Factor the number inside radical

For $\sqrt[3]{160}$, factor 160 as $64\times2.5$. Then $\sqrt[3]{160}=\sqrt[3]{64\times2.5}=\sqrt[3]{64}\times\sqrt[3]{2.5}=4\sqrt[3]{2.5}$.

Step8: Factor the number inside radical

For $\sqrt{45}$, factor 45 as $9\times5$. Then $\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}$.

Step9: Factor the number inside radical

For $\sqrt[3]{54}$, factor 54 as $27\times2$. Then $\sqrt[3]{54}=\sqrt[3]{27\times2}=\sqrt[3]{27}\times\sqrt[3]{2}=3\sqrt[3]{2}$.

Answer:

1. $2\sqrt{10}$
2. $5\sqrt{2}$
3. $2\sqrt[3]{2}$
4. $6\sqrt{2}$
5. $3$
6. $4\sqrt{2}$
7. $4\sqrt[3]{2.5}$
8. $3\sqrt{5}$
9. $3\sqrt[3]{2}$