Solve 64/128-16sqrt{2}

$$\frac{ 64 }{ 128-16 \sqrt{ 2 } }$$

Evaluate

$$\frac{2\left(\sqrt{2}+8\right)}{31}\approx 0.607368617$$

Solution Steps

Rationalize the denominator of \frac{64}{128-16\sqrt{2}} by multiplying numerator and denominator by 128+16\sqrt{2}.

$$\frac{64\left(128+16\sqrt{2}\right)}{\left(128-16\sqrt{2}\right)\left(128+16\sqrt{2}\right)}$$

Consider \left(128-16\sqrt{2}\right)\left(128+16\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}.

$$\frac{64\left(128+16\sqrt{2}\right)}{128^{2}-\left(-16\sqrt{2}\right)^{2}}$$

Calculate 128 to the power of 2 and get 16384.

$$\frac{64\left(128+16\sqrt{2}\right)}{16384-\left(-16\sqrt{2}\right)^{2}}$$

Expand \left(-16\sqrt{2}\right)^{2}.

$$\frac{64\left(128+16\sqrt{2}\right)}{16384-\left(-16\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -16 to the power of 2 and get 256.

$$\frac{64\left(128+16\sqrt{2}\right)}{16384-256\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{64\left(128+16\sqrt{2}\right)}{16384-256\times 2}$$

Multiply 256 and 2 to get 512.

$$\frac{64\left(128+16\sqrt{2}\right)}{16384-512}$$

Subtract 512 from 16384 to get 15872.

$$\frac{64\left(128+16\sqrt{2}\right)}{15872}$$

Divide 64\left(128+16\sqrt{2}\right) by 15872 to get \frac{1}{248}\left(128+16\sqrt{2}\right).

$$\frac{1}{248}\left(128+16\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{248} by 128+16\sqrt{2}.

$$\frac{1}{248}\times 128+\frac{1}{248}\times 16\sqrt{2}$$

Multiply \frac{1}{248} and 128 to get \frac{128}{248}.

$$\frac{128}{248}+\frac{1}{248}\times 16\sqrt{2}$$

Reduce the fraction \frac{128}{248} to lowest terms by extracting and canceling out 8.

$$\frac{16}{31}+\frac{1}{248}\times 16\sqrt{2}$$

Multiply \frac{1}{248} and 16 to get \frac{16}{248}.

$$\frac{16}{31}+\frac{16}{248}\sqrt{2}$$

Reduce the fraction \frac{16}{248} to lowest terms by extracting and canceling out 8.

$$\frac{16}{31}+\frac{2}{31}\sqrt{2}$$