What is the answer to 16=((2^{3n+1}+2^{n+2}))/((2+4^n)) ?

The answer to 16=((2^{3n+1}+2^{n+2}))/((2+4^n)) is n=3

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16=((2^{3n+1}+2^{n+2}))/((2+4^n))

16 = \frac{( 2 ^{3 n + 1} + 2 ^{n + 2} )}{( 2 + 4 ^{n} )}

n = 3

Solution steps

16 = \frac{( 2 ^{3 n + 1} + 2 ^{n + 2} )}{( 2 + 4 ^{n} )}

Multiply both sides by

2 + 4^{n}

16 (2 + 4^{n}) = \frac{2 ^{3 n + 1} + 2 ^{n + 2}}{2 + 4 ^{n}} (2 + 4^{n})

Simplify

16 (2 + 4^{n}) = 2^{3 n + 1} + 2^{n + 2}

Apply exponent rules

16 (2 + (2^{n})^{2}) = (2^{n})^{3} \cdot 2^{1} + 2^{n} \cdot 2^{2}

Rewrite the equation with

2^{n} = u

16 (2 + (u)^{2}) = (u)^{3} \cdot 2^{1} + u \cdot 2^{2}

Solve

16 (2 + u^{2}) = u^{3} \cdot 2^{1} + u \cdot 2^{2} : u = 8

u = 8

Substitute back

u = 2^{n},

solve for

n

Solve

2^{n} = 8 : n = 3

n = 3

Verify Solutions:

n = 3

True

The solution is

n = 3

5^{n + 1} + 5^{2 - n} = 5^{3} + 1

e^{2} x - 6 e^{x} + 5 = 0

4^{- x} = (2^{- x})^{2}

2 5^{n - 1} + 100 = 5^{2 n - 1}

1 6^{x} = 4

What is the answer to 16=((2^{3n+1}+2^{n+2}))/((2+4^n)) ?
The answer to 16=((2^{3n+1}+2^{n+2}))/((2+4^n)) is n=3
Solve for n: 16=((2^{3n+1}+2^{n+2}))/((2+4^n))
The solution is n=3