If x is a positive integer such that \frac{4^x + 4^x + 4^x + 4^x}{2^x + 2^x} = 512 what is… — GRE General Test

x

is a positive integer such that

\frac{4^x + 4^x + 4^x + 4^x}{2^x + 2^x} = 512

what is the value of

x

Answer: 8

The value of

x

is 8.

Combining four terms of

4^x

yields

4 \cdot 4^x = 4^{x+1} = 2^{2x+2}

. Combining two terms of

2^x

yields

2 \cdot 2^x = 2^{x+1}

. Dividing the numerator by the denominator gives

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

. Since

512 = 2^9

, setting

2^{x+1} = 2^9

gives

x + 1 = 9

, which leads directly to

x = 8

Simplify the numerator by combining identical added terms.

The numerator

4^x + 4^x + 4^x + 4^x

equals

4 \cdot 4^x

, which simplifies to

4^{x+1}

Adding four identical quantities is equivalent to multiplying that quantity by 4.

Simplify the denominator by combining identical added terms.

The denominator

2^x + 2^x

equals

2 \cdot 2^x

, which simplifies to

2^{x+1}

Adding two identical quantities is equivalent to multiplying that quantity by 2.

Convert the numerator to base 2 and simplify the fraction.

Since

4^{x+1} = (2^2)^{x+1} = 2^{2x+2}

, the fraction becomes

\frac{2^{2x+2}}{2^{x+1}} = 2^{(2x+2)-(x+1)} = 2^{x+1}

Converting all terms to a common base allows using the exponent quotient rule

a^m / a^n = a^{m-n}

Solve for

x

by equating the simplified power to 512.

Setting

2^{x+1} = 512 = 2^9

yields

x + 1 = 9

, so

x = 8

When exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

Combining repeated addition of exponential terms and converting powers to a common base using exponent laws (

a^m \cdot a^n = a^{m+n}

and

\frac{a^m}{a^n} = a^{m-n}

Question