(0.1 × 0.1 × 0.1 + 0.04 × 0.04 × 0.04) ÷ (0.2 × 0.2 × 0.2 + 0.08 × 0.08 × 0.08) = ______.
0.125
Solving the Decimal Expression
Let's solve the given mathematical expression step by step. The expression is:
\((0.1 \times 0.1 \times 0.1 + 0.04 \times 0.04 \times 0.04) \div (0.2 \times 0.2 \times 0.2 + 0.08 \times 0.08 \times 0.08)\)
We can rewrite the terms using exponents:
\((0.1^3 + 0.04^3) \div (0.2^3 + 0.08^3)\)
Let's look at the terms in the denominator and compare them to the terms in the numerator:
- \(0.2 = 2 \times 0.1\)
- \(0.08 = 2 \times 0.04\)
Now substitute these into the denominator:
\((2 \times 0.1)^3 + (2 \times 0.04)^3\)
Using the exponent rule \((ab)^n = a^n b^n\), we can write:
\(2^3 \times 0.1^3 + 2^3 \times 0.04^3\)
We can factor out \(2^3\) from both terms in the denominator:
\(2^3 (0.1^3 + 0.04^3)\)
Since \(2^3 = 2 \times 2 \times 2 = 8\), the denominator is:
\(8 (0.1^3 + 0.04^3)\)
Now substitute this back into the original expression:
\(\frac{0.1^3 + 0.04^3}{8 (0.1^3 + 0.04^3)}\)
We can see that the term \((0.1^3 + 0.04^3)\) appears in both the numerator and the denominator. As \(0.1^3 = 0.001\) and \(0.04^3 = 0.000064\), their sum is \(0.001064\), which is not zero. Therefore, we can cancel out this common term:
\(\frac{\cancel{(0.1^3 + 0.04^3)}}{8 \cancel{(0.1^3 + 0.04^3)}}\)
This simplifies the expression to:
\(\frac{1}{8}\)
Finally, convert the fraction \(\frac{1}{8}\) to a decimal:
\(1 \div 8 = 0.125\)
So, the value of the expression is 0.125.
Revision Table: Key Calculation Steps
| Step | Calculation | Result |
|---|---|---|
| Numerator terms | \(0.1^3\) | \(0.001\) |
| Numerator terms | \(0.04^3\) | \(0.000064\) |
| Denominator term relationship | \(0.2 = 2 \times 0.1\) | - |
| Denominator term relationship | \(0.08 = 2 \times 0.04\) | - |
| Denominator using relationship | \((2 \times 0.1)^3 + (2 \times 0.04)^3\) | \(2^3(0.1^3 + 0.04^3)\) |
| Simplify fraction | \(\frac{0.1^3 + 0.04^3}{8 (0.1^3 + 0.04^3)}\) | \(\frac{1}{8}\) |
| Convert to decimal | \(\frac{1}{8}\) | \(0.125\) |
Additional Information: Properties of Exponents and Decimals
This problem utilizes basic arithmetic operations with decimals and properties of exponents. Understanding these concepts is crucial for solving such expressions.
- Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, \(a^3 = a \times a \times a\).
- Property \((ab)^n = a^n b^n\): When a product of numbers is raised to a power, you can raise each number in the product to that power separately and then multiply the results. This property was key to simplifying the denominator in the problem.
- Decimal Arithmetic: Performing multiplication and division with decimals requires careful handling of the decimal point. Converting fractions like \(\frac{1}{8}\) to decimals involves dividing the numerator by the denominator. \(\frac{1}{8}\) is a common fraction equivalent to 0.125.
Breaking down complex expressions into simpler parts and identifying relationships between terms can make solving them much easier.
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