What is the answer to 64^x=8^{x+1}-16 ?

The answer to 64^x=8^{x+1}-16 is x= 2/3

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64^x=8^{x+1}-16

6 4^{x} = 8^{x + 1} - 16

x = \frac{2}{3}

Decimal Notation

x = 0.66666 \dots

Solution steps

6 4^{x} = 8^{x + 1} - 16

Apply exponent rules

(8^{x})^{2} = 8^{x} \cdot 8^{1} - 16

Rewrite the equation with

8^{x} = u

(u)^{2} = u \cdot 8^{1} - 16

Solve

u^{2} = u \cdot 8^{1} - 16 : u = 4

u = 4

Substitute back

u = 8^{x},

solve for

x

Solve

8^{x} = 4 : x = \frac{2}{3}

x = \frac{2}{3}

2^{x + 1} + 2^{x} + 2^{x - 1} = 112

32 = 1 6^{x - 4}

1 0^{2 x} - 2 (1 0^{x}) + 1 = 0

3^{5 x - 1} = 4^{3 x + 6}

(\frac{1}{16})^{x} = 64