If 4^x + 4^{x+1} + 4^{x+2} = 84 \cdot 2^{10}, what is the value of x? — GMAT

If $4^x + 4^{x+1} + 4^{x+2} = 84 \cdot 2^{10}$ , what is the value of $x$ ?

Factoring out

4^x

gives

4^x(1 + 4 + 16) = 21 \cdot 4^x

. Dividing both sides of

21 \cdot 4^x = 84 \cdot 2^{10}

by 21 yields

4^x = 4 \cdot 2^{10}

. Converting terms to base 2 produces

2^{2x} = 2^2 \cdot 2^{10} = 2^{12}

. Setting exponents equal gives

2x = 12

, which yields

x = 6

Factor out the common term

4^x

from the left side of the equation.

4^x(1 + 4^1 + 4^2) = 4^x(1 + 4 + 16) = 21 \cdot 4^x

Expressions adding powers with consecutive exponents can be simplified by factoring out the lowest power.

Substitute the factored expression into the equation and solve for

4^x

21 \cdot 4^x = 84 \cdot 2^{10} \implies 4^x = \frac{84}{21} \cdot 2^{10} = 4 \cdot 2^{10}

Dividing both sides by 21 isolates the exponential term on the left.

Convert both sides of the equation to powers of base 2.

(2^2)^x = 2^2 \cdot 2^{10} \implies 2^{2x} = 2^{12}

Expressing both sides with a common base enables equating exponents.

Equate exponents and solve for

x

2x = 12 \implies x = 6

When bases are equal, the powers must be equal.

Factoring Exponential Expressions and Base Conversion

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