Solve 384*10^8/sqrt{frac{598{735}}+1}

$$\frac { 384 \times 10 ^ { 8 } } { \sqrt { \frac { 598 } { 735 } } + 1 }$$

Evaluate

$$\frac{28224000000000-268800000000\sqrt{8970}}{137}\approx 20189265962.168567657$$

Solution Steps

Calculate 10 to the power of 8 and get 100000000.

$$\frac{384\times 100000000}{\sqrt{\frac{598}{735}}+1}$$

Multiply 384 and 100000000 to get 38400000000.

$$\frac{38400000000}{\sqrt{\frac{598}{735}}+1}$$

Rewrite the square root of the division \sqrt{\frac{598}{735}} as the division of square roots \frac{\sqrt{598}}{\sqrt{735}}.

$$\frac{38400000000}{\frac{\sqrt{598}}{\sqrt{735}}+1}$$

Factor 735=7^{2}\times 15. Rewrite the square root of the product \sqrt{7^{2}\times 15} as the product of square roots \sqrt{7^{2}}\sqrt{15}. Take the square root of 7^{2}.

$$\frac{38400000000}{\frac{\sqrt{598}}{7\sqrt{15}}+1}$$

Rationalize the denominator of \frac{\sqrt{598}}{7\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{38400000000}{\frac{\sqrt{598}\sqrt{15}}{7\left(\sqrt{15}\right)^{2}}+1}$$

The square of \sqrt{15} is 15.

$$\frac{38400000000}{\frac{\sqrt{598}\sqrt{15}}{7\times 15}+1}$$

To multiply \sqrt{598} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{38400000000}{\frac{\sqrt{8970}}{7\times 15}+1}$$

Multiply 7 and 15 to get 105.

$$\frac{38400000000}{\frac{\sqrt{8970}}{105}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{105}{105}.

$$\frac{38400000000}{\frac{\sqrt{8970}}{105}+\frac{105}{105}}$$

Since \frac{\sqrt{8970}}{105} and \frac{105}{105} have the same denominator, add them by adding their numerators.

$$\frac{38400000000}{\frac{\sqrt{8970}+105}{105}}$$

Divide 38400000000 by \frac{\sqrt{8970}+105}{105} by multiplying 38400000000 by the reciprocal of \frac{\sqrt{8970}+105}{105}.

$$\frac{38400000000\times 105}{\sqrt{8970}+105}$$

Rationalize the denominator of \frac{38400000000\times 105}{\sqrt{8970}+105} by multiplying numerator and denominator by \sqrt{8970}-105.

$$\frac{38400000000\times 105\left(\sqrt{8970}-105\right)}{\left(\sqrt{8970}+105\right)\left(\sqrt{8970}-105\right)}$$

Consider \left(\sqrt{8970}+105\right)\left(\sqrt{8970}-105\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{38400000000\times 105\left(\sqrt{8970}-105\right)}{\left(\sqrt{8970}\right)^{2}-105^{2}}$$

Square \sqrt{8970}. Square 105.

$$\frac{38400000000\times 105\left(\sqrt{8970}-105\right)}{8970-11025}$$

Subtract 11025 from 8970 to get -2055.

$$\frac{38400000000\times 105\left(\sqrt{8970}-105\right)}{-2055}$$

Multiply 38400000000 and 105 to get 4032000000000.

$$\frac{4032000000000\left(\sqrt{8970}-105\right)}{-2055}$$

Divide 4032000000000\left(\sqrt{8970}-105\right) by -2055 to get -\frac{268800000000}{137}\left(\sqrt{8970}-105\right).

$$-\frac{268800000000}{137}\left(\sqrt{8970}-105\right)$$

Use the distributive property to multiply -\frac{268800000000}{137} by \sqrt{8970}-105.

$$-\frac{268800000000}{137}\sqrt{8970}-\frac{268800000000}{137}\left(-105\right)$$

Express -\frac{268800000000}{137}\left(-105\right) as a single fraction.

$$-\frac{268800000000}{137}\sqrt{8970}+\frac{-268800000000\left(-105\right)}{137}$$

Multiply -268800000000 and -105 to get 28224000000000.

$$-\frac{268800000000}{137}\sqrt{8970}+\frac{28224000000000}{137}$$