Solve frac{4sqrt{10}+3sqrt{2}}{4sqrt{5}+3}

$$\frac { 4 \sqrt { 10 } + 3 \sqrt { 2 } } { 4 \sqrt { 5 } + 3 }$$

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

Rationalize the denominator of \frac{4\sqrt{10}+3\sqrt{2}}{4\sqrt{5}+3} by multiplying numerator and denominator by 4\sqrt{5}-3.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{\left(4\sqrt{5}+3\right)\left(4\sqrt{5}-3\right)}$$

Consider \left(4\sqrt{5}+3\right)\left(4\sqrt{5}-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{\left(4\sqrt{5}\right)^{2}-3^{2}}$$

Expand \left(4\sqrt{5}\right)^{2}.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{4^{2}\left(\sqrt{5}\right)^{2}-3^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{16\left(\sqrt{5}\right)^{2}-3^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{16\times 5-3^{2}}$$

Multiply 16 and 5 to get 80.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{80-3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{80-9}$$

Subtract 9 from 80 to get 71.

$$\frac{\left(4\sqrt{10}+3\sqrt{2}\right)\left(4\sqrt{5}-3\right)}{71}$$

Apply the distributive property by multiplying each term of 4\sqrt{10}+3\sqrt{2} by each term of 4\sqrt{5}-3.

$$\frac{16\sqrt{10}\sqrt{5}-12\sqrt{10}+12\sqrt{2}\sqrt{5}-9\sqrt{2}}{71}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{16\sqrt{5}\sqrt{2}\sqrt{5}-12\sqrt{10}+12\sqrt{2}\sqrt{5}-9\sqrt{2}}{71}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{16\times 5\sqrt{2}-12\sqrt{10}+12\sqrt{2}\sqrt{5}-9\sqrt{2}}{71}$$

Multiply 16 and 5 to get 80.

$$\frac{80\sqrt{2}-12\sqrt{10}+12\sqrt{2}\sqrt{5}-9\sqrt{2}}{71}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{80\sqrt{2}-12\sqrt{10}+12\sqrt{10}-9\sqrt{2}}{71}$$

Combine -12\sqrt{10} and 12\sqrt{10} to get 0.

$$\frac{80\sqrt{2}-9\sqrt{2}}{71}$$

Combine 80\sqrt{2} and -9\sqrt{2} to get 71\sqrt{2}.

$$\frac{71\sqrt{2}}{71}$$

Cancel out 71 and 71.

$$\sqrt{2}$$