If x is a positive integer such that \frac{6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6}{3^6 + 3^6 + … — GMAT

If $x$ is a positive integer such that $\frac{6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6}{3^6 + 3^6 + 3^6} = 2^x$ , what is the value of $x$ ?

Answer: 7

The value of

x

is 7.

Repeated addition converts to multiplication: six terms of

6^6

yield

6 \cdot 6^6 = 6^7

, and three terms of

3^6

yield

3 \cdot 3^6 = 3^7

. Dividing gives

\frac{6^7}{3^7} = \left(\frac{6}{3}\right)^7 = 2^7

. Comparing

2^7

2^x

gives

x = 7

Simplify the numerator by expressing repeated addition as multiplication.

6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = 6 \times 6^6 = 6^7

Adding six identical terms of

6^6

is equivalent to multiplying

6^6

by 6. Using the power rule

a^1 \cdot a^n = a^{n+1}

, we get

6^7

Simplify the denominator by expressing repeated addition as multiplication.

3^6 + 3^6 + 3^6 = 3 \times 3^6 = 3^7

Adding three identical terms of

3^6

is equivalent to multiplying

3^6

by 3, yielding

3^7

Apply the quotient property of exponents for identical powers.

\frac{6^7}{3^7} = \left(\frac{6}{3}\right)^7 = 2^7

According to exponent laws,

\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n

for any non-zero real numbers

a

and

b

Equate exponents of equal bases to solve for

x

2^x = 2^7 \implies x = 7

Since the bases on both sides of the equation are equal to 2, the exponents must be equal.

Combining repeated addition into exponential products and dividing powers with equal exponents.

Question