Simplify Mathematical Expression with Cubes
The question asks to simplify the following mathematical expression:
\( \frac{0.8 \times 0.8 \times 0.8 - 0.08 \times 0.08 \times 0.08}{8 \times 8 \times 8} \)Rewriting the Expression
First, rewrite the expression using exponential notation:
\( \frac{(0.8)^3 - (0.08)^3}{8^3} \)Applying Properties of Exponents
Notice that \(0.8 = 8 \times 10^{-1}\) and \(0.08 = 8 \times 10^{-2}\). Substitute these into the expression:
\( \frac{(8 \times 10^{-1})^3 - (8 \times 10^{-2})^3}{8^3} \)Using the property \((a \times b)^n = a^n \times b^n\):
\( \frac{8^3 \times (10^{-1})^3 - 8^3 \times (10^{-2})^3}{8^3} \)Using the property \((a^m)^n = a^{m \times n}\):
\( \frac{8^3 \times 10^{-3} - 8^3 \times 10^{-6}}{8^3} \)Factoring and Simplifying
Factor out \(8^3\) from the numerator:
\( \frac{8^3 (10^{-3} - 10^{-6})}{8^3} \)Cancel out the common factor \(8^3\):
\( 10^{-3} - 10^{-6} \)Final Calculation
Convert the negative exponents to decimals:
\( 10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001 \) \( 10^{-6} = \frac{1}{10^6} = \frac{1}{1000000} = 0.000001 \)Perform the subtraction:
\( 0.001 - 0.000001 = 0.000999 \)Conclusion
The simplified value of the expression is 0.000999.
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