Solve frac{sqrt{99}+1}{sqrt{98}-1}

$$\frac { \sqrt { 99 } + 1 } { \sqrt { 98 } - 1 }$$

Evaluate

$$\frac{3\sqrt{11}+7\sqrt{2}+21\sqrt{22}+1}{97}\approx 1.230392786$$

Solution Steps

Factor 99=3^{2}\times 11. Rewrite the square root of the product \sqrt{3^{2}\times 11} as the product of square roots \sqrt{3^{2}}\sqrt{11}. Take the square root of 3^{2}.

$$\frac{3\sqrt{11}+1}{\sqrt{98}-1}$$

Factor 98=7^{2}\times 2. Rewrite the square root of the product \sqrt{7^{2}\times 2} as the product of square roots \sqrt{7^{2}}\sqrt{2}. Take the square root of 7^{2}.

$$\frac{3\sqrt{11}+1}{7\sqrt{2}-1}$$

Rationalize the denominator of \frac{3\sqrt{11}+1}{7\sqrt{2}-1} by multiplying numerator and denominator by 7\sqrt{2}+1.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{\left(7\sqrt{2}-1\right)\left(7\sqrt{2}+1\right)}$$

Consider \left(7\sqrt{2}-1\right)\left(7\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{\left(7\sqrt{2}\right)^{2}-1^{2}}$$

Expand \left(7\sqrt{2}\right)^{2}.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{7^{2}\left(\sqrt{2}\right)^{2}-1^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{49\left(\sqrt{2}\right)^{2}-1^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{49\times 2-1^{2}}$$

Multiply 49 and 2 to get 98.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{98-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{98-1}$$

Subtract 1 from 98 to get 97.

$$\frac{\left(3\sqrt{11}+1\right)\left(7\sqrt{2}+1\right)}{97}$$

Apply the distributive property by multiplying each term of 3\sqrt{11}+1 by each term of 7\sqrt{2}+1.

$$\frac{21\sqrt{11}\sqrt{2}+3\sqrt{11}+7\sqrt{2}+1}{97}$$

To multiply \sqrt{11} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{21\sqrt{22}+3\sqrt{11}+7\sqrt{2}+1}{97}$$