What is the answer to 5^{n+1}+5^{2-n}=5^3+1 ?

The answer to 5^{n+1}+5^{2-n}=5^3+1 is n=2,n=-1

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

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

n = 2, n = - 1

Solution steps

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

Apply exponent rules

5^{n} \cdot 5^{1} + (5^{n})^{- 1} \cdot 5^{2} = 5^{3} + 1

Rewrite the equation with

5^{n} = u

u \cdot 5^{1} + (u)^{- 1} \cdot 5^{2} = 5^{3} + 1

Solve

u \cdot 5^{1} + u^{- 1} \cdot 5^{2} = 5^{3} + 1 : u = 25, u = \frac{1}{5}

u = 25, u = \frac{1}{5}

Substitute back

u = 5^{n},

solve for

n

Solve

5^{n} = 25 : n = 2

Solve

5^{n} = \frac{1}{5} : n = - 1

n = 2, n = - 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

4^{3 t} = 8^{t - 1}

What is the answer to 5^{n+1}+5^{2-n}=5^3+1 ?
The answer to 5^{n+1}+5^{2-n}=5^3+1 is n=2,n=-1
Solve for n: 5^{n+1}+5^{2-n}=5^3+1
The solution is n=2,n=-1