Solve 1/3sqrt{7+8}

$$\frac { 1 } { 3 \sqrt { 7 } + 8 }$$

Evaluate

$$8-3\sqrt{7}\approx 0.062746067$$

Solution Steps

Rationalize the denominator of \frac{1}{3\sqrt{7}+8} by multiplying numerator and denominator by 3\sqrt{7}-8.

$$\frac{3\sqrt{7}-8}{\left(3\sqrt{7}+8\right)\left(3\sqrt{7}-8\right)}$$

Consider \left(3\sqrt{7}+8\right)\left(3\sqrt{7}-8\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\sqrt{7}-8}{\left(3\sqrt{7}\right)^{2}-8^{2}}$$

Expand \left(3\sqrt{7}\right)^{2}.

$$\frac{3\sqrt{7}-8}{3^{2}\left(\sqrt{7}\right)^{2}-8^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3\sqrt{7}-8}{9\left(\sqrt{7}\right)^{2}-8^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{3\sqrt{7}-8}{9\times 7-8^{2}}$$

Multiply 9 and 7 to get 63.

$$\frac{3\sqrt{7}-8}{63-8^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{3\sqrt{7}-8}{63-64}$$

Subtract 64 from 63 to get -1.

$$\frac{3\sqrt{7}-8}{-1}$$

Anything divided by -1 gives its opposite. To find the opposite of 3\sqrt{7}-8, find the opposite of each term.

$$-3\sqrt{7}-\left(-8\right)$$

The opposite of -8 is 8.

$$-3\sqrt{7}+8$$