Solve 6/3sqrt{2+4^2}=

$$\frac { 6 } { 3 \sqrt { 2 } + 4 ^ { 2 } } =$$

Evaluate

$$\frac{48-9\sqrt{2}}{119}\approx 0.296404016$$

Solution Steps

Calculate 4 to the power of 2 and get 16.

$$\frac{6}{3\sqrt{2}+16}$$

Rationalize the denominator of \frac{6}{3\sqrt{2}+16} by multiplying numerator and denominator by 3\sqrt{2}-16.

$$\frac{6\left(3\sqrt{2}-16\right)}{\left(3\sqrt{2}+16\right)\left(3\sqrt{2}-16\right)}$$

Consider \left(3\sqrt{2}+16\right)\left(3\sqrt{2}-16\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{6\left(3\sqrt{2}-16\right)}{\left(3\sqrt{2}\right)^{2}-16^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{6\left(3\sqrt{2}-16\right)}{3^{2}\left(\sqrt{2}\right)^{2}-16^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{6\left(3\sqrt{2}-16\right)}{9\left(\sqrt{2}\right)^{2}-16^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{6\left(3\sqrt{2}-16\right)}{9\times 2-16^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{6\left(3\sqrt{2}-16\right)}{18-16^{2}}$$

Calculate 16 to the power of 2 and get 256.

$$\frac{6\left(3\sqrt{2}-16\right)}{18-256}$$

Subtract 256 from 18 to get -238.

$$\frac{6\left(3\sqrt{2}-16\right)}{-238}$$

Divide 6\left(3\sqrt{2}-16\right) by -238 to get -\frac{3}{119}\left(3\sqrt{2}-16\right).

$$-\frac{3}{119}\left(3\sqrt{2}-16\right)$$

Use the distributive property to multiply -\frac{3}{119} by 3\sqrt{2}-16.

$$-\frac{3}{119}\times 3\sqrt{2}-\frac{3}{119}\left(-16\right)$$

Express -\frac{3}{119}\times 3 as a single fraction.

$$\frac{-3\times 3}{119}\sqrt{2}-\frac{3}{119}\left(-16\right)$$

Multiply -3 and 3 to get -9.

$$\frac{-9}{119}\sqrt{2}-\frac{3}{119}\left(-16\right)$$

Fraction \frac{-9}{119} can be rewritten as -\frac{9}{119} by extracting the negative sign.

$$-\frac{9}{119}\sqrt{2}-\frac{3}{119}\left(-16\right)$$

Express -\frac{3}{119}\left(-16\right) as a single fraction.

$$-\frac{9}{119}\sqrt{2}+\frac{-3\left(-16\right)}{119}$$

Multiply -3 and -16 to get 48.

$$-\frac{9}{119}\sqrt{2}+\frac{48}{119}$$