Expression Simplification Calculation

The expression to simplify is: $9 \times 0.9 \times 0.09 \times 0.009 \times \frac{1}{0.3} \times \frac{1}{0.03} \times \frac{1}{0.003}$

Step 1: Rewrite terms using powers.

Express each number in terms of base 9, base $\frac{1}{3}$, and powers of 10:

• $9 = 9 \times 10^0$
• $0.9 = 9 \times 10^{-1}$
• $0.09 = 9 \times 10^{-2}$
• $0.009 = 9 \times 10^{-3}$
• $\frac{1}{0.3} = \frac{1}{3 \times 10^{-1}} = \frac{1}{3} \times 10^1$
• $\frac{1}{0.03} = \frac{1}{3 \times 10^{-2}} = \frac{1}{3} \times 10^2$
• $\frac{1}{0.003} = \frac{1}{3 \times 10^{-3}} = \frac{1}{3} \times 10^3$

Step 2: Group similar bases.

Substitute these rewritten terms back into the original expression and group coefficients and powers of 10 separately:

Expression = $(9 \times 9 \times 9 \times 9) \times (\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}) \times (10^0 \times 10^{-1} \times 10^{-2} \times 10^{-3} \times 10^1 \times 10^2 \times 10^3)$

Step 3: Calculate each group.

Calculate the product for each group:

• Product of base 9 terms: $9^4 = 6561$
• Product of base $\frac{1}{3}$ terms: $(\frac{1}{3})^3 = \frac{1}{27}$
• Product of base 10 terms (sum exponents): $10^{(0 - 1 - 2 - 3 + 1 + 2 + 3)} = 10^0 = 1$

Step 4: Multiply the results.

Multiply the results from Step 3 to find the final simplified value:

Result = $6561 \times \frac{1}{27} \times 1 = \frac{6561}{27}$

Result = $243$

The simplified value of the expression is 243.