Solve 6sqrt{2}-6+12/10+6sqrt{2}

$$6 \sqrt { 2 } - 6 + \frac { 12 } { 10 + 6 \sqrt { 2 } }$$

Evaluate

$$\frac{24\sqrt{2}-12}{7}\approx 3.1344465$$

Solution Steps

Rationalize the denominator of \frac{12}{10+6\sqrt{2}} by multiplying numerator and denominator by 10-6\sqrt{2}.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{\left(10+6\sqrt{2}\right)\left(10-6\sqrt{2}\right)}$$

Consider \left(10+6\sqrt{2}\right)\left(10-6\sqrt{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}.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{10^{2}-\left(6\sqrt{2}\right)^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{100-\left(6\sqrt{2}\right)^{2}}$$

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

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{100-6^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{100-36\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{100-36\times 2}$$

Multiply 36 and 2 to get 72.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{100-72}$$

Subtract 72 from 100 to get 28.

$$6\sqrt{2}-6+\frac{12\left(10-6\sqrt{2}\right)}{28}$$

Divide 12\left(10-6\sqrt{2}\right) by 28 to get \frac{3}{7}\left(10-6\sqrt{2}\right).

$$6\sqrt{2}-6+\frac{3}{7}\left(10-6\sqrt{2}\right)$$

Use the distributive property to multiply \frac{3}{7} by 10-6\sqrt{2}.

$$6\sqrt{2}-6+\frac{3}{7}\times 10+\frac{3}{7}\left(-6\right)\sqrt{2}$$

Express \frac{3}{7}\times 10 as a single fraction.

$$6\sqrt{2}-6+\frac{3\times 10}{7}+\frac{3}{7}\left(-6\right)\sqrt{2}$$

Multiply 3 and 10 to get 30.

$$6\sqrt{2}-6+\frac{30}{7}+\frac{3}{7}\left(-6\right)\sqrt{2}$$

Express \frac{3}{7}\left(-6\right) as a single fraction.

$$6\sqrt{2}-6+\frac{30}{7}+\frac{3\left(-6\right)}{7}\sqrt{2}$$

Multiply 3 and -6 to get -18.

$$6\sqrt{2}-6+\frac{30}{7}+\frac{-18}{7}\sqrt{2}$$

Fraction \frac{-18}{7} can be rewritten as -\frac{18}{7} by extracting the negative sign.

$$6\sqrt{2}-6+\frac{30}{7}-\frac{18}{7}\sqrt{2}$$

Convert -6 to fraction -\frac{42}{7}.

$$6\sqrt{2}-\frac{42}{7}+\frac{30}{7}-\frac{18}{7}\sqrt{2}$$

Since -\frac{42}{7} and \frac{30}{7} have the same denominator, add them by adding their numerators.

$$6\sqrt{2}+\frac{-42+30}{7}-\frac{18}{7}\sqrt{2}$$

Add -42 and 30 to get -12.

$$6\sqrt{2}-\frac{12}{7}-\frac{18}{7}\sqrt{2}$$

Combine 6\sqrt{2} and -\frac{18}{7}\sqrt{2} to get \frac{24}{7}\sqrt{2}.

$$\frac{24}{7}\sqrt{2}-\frac{12}{7}$$

Factor

$$\frac{12 {(2 \sqrt{2} - 1)}}{7} = 3.134446499564898$$