Here are the solutions for each problem on the worksheet. To rationalize a denominator, we multiply the top and bottom of the fraction by the square root in the denominator. This turns the bottom number into a whole number. Then, we simplify the fraction if possible.
2. $\frac{2}{\sqrt{3}}$
* Multiply top and bottom by $\sqrt{3}$: $\frac{2 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{2\sqrt{3}}{3}$
* The fraction cannot be simplified further.
3. $\frac{1}{\sqrt{7}}$
* Multiply top and bottom by $\sqrt{7}$: $\frac{1 \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} = \frac{\sqrt{7}}{7}$
4. $\frac{6}{\sqrt{2}}$
* Multiply top and bottom by $\sqrt{2}$: $\frac{6 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{6\sqrt{2}}{2}$
* Simplify by dividing 6 by 2: $3\sqrt{2}$
5. $\frac{15}{\sqrt{5}}$
* Multiply top and bottom by $\sqrt{5}$: $\frac{15 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{15\sqrt{5}}{5}$
* Simplify by dividing 15 by 5: $3\sqrt{5}$
6. $\frac{42}{\sqrt{7}}$
* Multiply top and bottom by $\sqrt{7}$: $\frac{42 \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} = \frac{42\sqrt{7}}{7}$
* Simplify by dividing 42 by 7: $6\sqrt{7}$
7. $\frac{1}{\sqrt{81}}$
* First, notice that $\sqrt{81}$ is a perfect square. $\sqrt{81} = 9$.
* So the fraction is simply $\frac{1}{9}$.
8. $\frac{2}{\sqrt{11}}$
* Multiply top and bottom by $\sqrt{11}$: $\frac{2 \cdot \sqrt{11}}{\sqrt{11} \cdot \sqrt{11}} = \frac{2\sqrt{11}}{11}$
9. $\frac{4}{\sqrt{2}}$
* Multiply top and bottom by $\sqrt{2}$: $\frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{4\sqrt{2}}{2}$
* Simplify by dividing 4 by 2: $2\sqrt{2}$
10. $\frac{1}{\sqrt{3}}$
* Multiply top and bottom by $\sqrt{3}$: $\frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}$
11. $\frac{1}{\sqrt{225}}$
* First, notice that $\sqrt{225}$ is a perfect square. $\sqrt{225} = 15$.
* So the fraction is simply $\frac{1}{15}$.
12. $\frac{1}{3\sqrt{16}}$
* First, simplify the square root in the denominator. $\sqrt{16} = 4$.
* The denominator becomes $3 \cdot 4 = 12$.
* So the fraction is simply $\frac{1}{12}$.
Final Answer:
2. $\frac{2\sqrt{3}}{3}$
3. $\frac{\sqrt{7}}{7}$
4. $3\sqrt{2}$
5. $3\sqrt{5}$
6. $6\sqrt{7}$
7. $\frac{1}{9}$
8. $\frac{2\sqrt{11}}{11}$
9. $2\sqrt{2}$
10. $\frac{\sqrt{3}}{3}$
11. $\frac{1}{15}$
12. $\frac{1}{12}$
Parent Tip: Review the logic above to help your child master the concept of rationalizing the denominator worksheet.
