Evaluate (2^4+2^-4)/(4^4+1)

$\frac{2^{4} + 2^{- 4}}{4^{4} + 1}$

Step 1

Simplify the numerator.

Raise $2$ to the power of $4$ .

$\frac{16 + 2^{- 4}}{4^{4} + 1}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{16 + \frac{1}{2^{4}}}{4^{4} + 1}$

Raise $2$ to the power of $4$ .

$\frac{16 + \frac{1}{16}}{4^{4} + 1}$

To write $16$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{16 \cdot \frac{16}{16} + \frac{1}{16}}{4^{4} + 1}$

Combine $16$ and $\frac{16}{16}$ .

$\frac{\frac{16 \cdot 16}{16} + \frac{1}{16}}{4^{4} + 1}$

Combine the numerators over the common denominator.

$\frac{\frac{16 \cdot 16 + 1}{16}}{4^{4} + 1}$

Simplify the numerator.

Multiply $16$ by $16$ .

$\frac{\frac{256 + 1}{16}}{4^{4} + 1}$

Add $256$ and $1$ .

$\frac{\frac{257}{16}}{4^{4} + 1}$

Step 2

Simplify the denominator.

Raise $4$ to the power of $4$ .

$\frac{\frac{257}{16}}{256 + 1}$

Add $256$ and $1$ .

$\frac{\frac{257}{16}}{257}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{257}{16} \cdot \frac{1}{257}$

Step 4

Cancel the common factor of $257$ .

Cancel the common factor.

$\frac{257}{16} \cdot \frac{1}{257}$

Rewrite the expression.

$\frac{1}{16}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{16}$

Decimal Form:

$0.0625$