81 Infinite Product Evaluation

The problem asks us to find the value of an infinite product expression:

$\left(81\right)^{1/3} \times \left(81\right)^{1/9} \times \left(81\right)^{1/27} \times \left(81\right)^{1/81} \times \dots$

We can use the property of exponents that states $a^m \times a^n = a^{m+n}$. Applying this property to the given infinite product, we can combine the exponents since the base is the same (81):

$\left(81\right)^{\left(1/3 + 1/9 + 1/27 + 1/81 + \dots\right)}$

The exponents form an infinite series: $1/3 + 1/9 + 1/27 + 1/81 + \dots$. This is a geometric series.

Geometric Series Sum

A geometric series is a series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

In the series $1/3 + 1/9 + 1/27 + 1/81 + \dots$:

• The first term, $a$, is $1/3$.
• The common ratio, $r$, is found by dividing any term by its preceding term. For example, $(1/9) / (1/3) = 1/9 \times 3/1 = 3/9 = 1/3$. Also, $(1/27) / (1/9) = 1/27 \times 9/1 = 9/27 = 1/3$. So, the common ratio $r = 1/3$.

Since the absolute value of the common ratio, $|r| = |1/3| = 1/3$, is less than 1, the sum of this infinite geometric series converges to a finite value. The formula for the sum $S$ of an infinite geometric series with first term $a$ and common ratio $r$ (where $|r| < 1$) is:

$S = \frac{a}{1-r}$

Substituting the values $a = 1/3$ and $r = 1/3$ into the formula:

$S = \frac{1/3}{1 - 1/3} = \frac{1/3}{2/3}$

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:

$S = \frac{1}{3} \times \frac{3}{2} = \frac{3}{6} = \frac{1}{2}$

So, the sum of the infinite series of exponents is $1/2$.

Evaluating the 81 Expression

Now we can substitute the sum of the exponents back into the expression:

$\left(81\right)^{\left(1/3 + 1/9 + 1/27 + 1/81 + \dots\right)} = \left(81\right)^{1/2}$

The exponent $1/2$ indicates the square root. So, we need to find the square root of 81.

$81^{1/2} = \sqrt{81}$

We know that $9 \times 9 = 81$. Therefore, the square root of 81 is 9.

$\sqrt{81} = 9$

Thus, the value of the given infinite product expression is 9.

Final Answer Confirmation

The value of the expression $81^{1/3} \times 81^{1/9} \times 81^{1/27} \times 81^{1/81} \times \dots$ up to infinite term is 9.

Comparing this result with the given options, we find that 9 is one of the options.