Solve 10/frac{6{5}sqrt{5}-1.2}

$$\frac { 10 } { \frac { 6 } { 5 } \sqrt { 5 } - 1.2 }$$

Evaluate

$$\frac{25\left(\sqrt{5}+1\right)}{12}\approx 6.741808286$$

Solution Steps

Rationalize the denominator of \frac{10}{\frac{6}{5}\sqrt{5}-1.2} by multiplying numerator and denominator by \frac{6}{5}\sqrt{5}+1.2.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\left(\frac{6}{5}\sqrt{5}-1.2\right)\left(\frac{6}{5}\sqrt{5}+1.2\right)}$$

Consider \left(\frac{6}{5}\sqrt{5}-1.2\right)\left(\frac{6}{5}\sqrt{5}+1.2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\left(\frac{6}{5}\sqrt{5}\right)^{2}-1.2^{2}}$$

Expand \left(\frac{6}{5}\sqrt{5}\right)^{2}.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\left(\frac{6}{5}\right)^{2}\left(\sqrt{5}\right)^{2}-1.2^{2}}$$

Calculate \frac{6}{5} to the power of 2 and get \frac{36}{25}.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{36}{25}\left(\sqrt{5}\right)^{2}-1.2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{36}{25}\times 5-1.2^{2}}$$

Express \frac{36}{25}\times 5 as a single fraction.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{36\times 5}{25}-1.2^{2}}$$

Multiply 36 and 5 to get 180.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{180}{25}-1.2^{2}}$$

Reduce the fraction \frac{180}{25} to lowest terms by extracting and canceling out 5.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{36}{5}-1.2^{2}}$$

Calculate 1.2 to the power of 2 and get \frac{36}{25}.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{36}{5}-\frac{36}{25}}$$

Least common multiple of 5 and 25 is 25. Convert \frac{36}{5} and \frac{36}{25} to fractions with denominator 25.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{180}{25}-\frac{36}{25}}$$

Since \frac{180}{25} and \frac{36}{25} have the same denominator, subtract them by subtracting their numerators.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{180-36}{25}}$$

Subtract 36 from 180 to get 144.

$$\frac{10\left(\frac{6}{5}\sqrt{5}+1.2\right)}{\frac{144}{25}}$$

Divide 10\left(\frac{6}{5}\sqrt{5}+1.2\right) by \frac{144}{25} to get \frac{125}{72}\left(\frac{6}{5}\sqrt{5}+1.2\right).

$$\frac{125}{72}\left(\frac{6}{5}\sqrt{5}+1.2\right)$$

Use the distributive property to multiply \frac{125}{72} by \frac{6}{5}\sqrt{5}+1.2.

$$\frac{125}{72}\times \frac{6}{5}\sqrt{5}+\frac{125}{72}\times 1.2$$

Multiply \frac{125}{72} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{125\times 6}{72\times 5}\sqrt{5}+\frac{125}{72}\times 1.2$$

Do the multiplications in the fraction \frac{125\times 6}{72\times 5}.

$$\frac{750}{360}\sqrt{5}+\frac{125}{72}\times 1.2$$

Reduce the fraction \frac{750}{360} to lowest terms by extracting and canceling out 30.

$$\frac{25}{12}\sqrt{5}+\frac{125}{72}\times 1.2$$

Convert decimal number 1.2 to fraction \frac{12}{10}. Reduce the fraction \frac{12}{10} to lowest terms by extracting and canceling out 2.

$$\frac{25}{12}\sqrt{5}+\frac{125}{72}\times \frac{6}{5}$$

Multiply \frac{125}{72} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25}{12}\sqrt{5}+\frac{125\times 6}{72\times 5}$$

Do the multiplications in the fraction \frac{125\times 6}{72\times 5}.

$$\frac{25}{12}\sqrt{5}+\frac{750}{360}$$

Reduce the fraction \frac{750}{360} to lowest terms by extracting and canceling out 30.

$$\frac{25}{12}\sqrt{5}+\frac{25}{12}$$