If 3^x + 3^{x+1} + 3^{x+2} = 351, what is the value of (x + 1)^x? — GMAT

If $3^x + 3^{x+1} + 3^{x+2} = 351$ , what is the value of $(x + 1)^x$ ?

The value of

(x + 1)^x

64

Factoring

3^x

out of the sum gives

3^x(1 + 3 + 9) = 13 \cdot 3^x = 351

. Dividing

351

13

gives

3^x = 27

, so

x = 3

. Substituting

x = 3

into

(x + 1)^x

yields

(3 + 1)^3 = 4^3 = 64

Factor out the common exponential term

3^x

from the left side of the equation.

3^x(1 + 3^1 + 3^2) = 351

, which simplifies to

3^x(1 + 3 + 9) = 351

13 \cdot 3^x = 351

When adding terms with identical bases and variable exponents, factor out the term with the smallest exponent.

Solve for

3^x

and determine the value of

x

3^x = \frac{351}{13} = 27

. Since

27 = 3^3

x = 3

Divide both sides by

13

and set the powers of equal bases equal to find

x

Substitute

x = 3

into the target expression

(x + 1)^x

(3 + 1)^3 = 4^3 = 64

Evaluate the expression as requested by the question stem.

Factoring exponential expressions with common bases and applying exponent rules

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