What is the value of $\frac{(0.2\times 0.2\times 0.2+0.04\times 0.04\times 0.04)}{(0.4\times 0.4\times 0.4+0.08\times 0.08\times 0.08)}$ ?

Correct Answer: 0.125

Step-by-Step Explanation:

Step 1: Analyze the given expression

We need to find the value of the following fraction:

$\frac{(0.2\times 0.2\times 0.2+0.04\times 0.04\times 0.04)}{(0.4\times 0.4\times 0.4+0.08\times 0.08\times 0.08)}$

Step 2: Identify the algebraic pattern between numerator and denominator

Observe the relationship between the numbers in the numerator and denominator:

$0.4=2\times 0.2$

$0.08=2\times 0.04$

Step 3: Express the terms in the denominator in terms of the numerator

Using the exponent power rule ${(a\times b)}^3=a^3\times b^3$:

$0.4\times 0.4\times 0.4={(2\times 0.2)}^3=2^3\times (0.2\times 0.2\times 0.2)=8\times (0.2\times 0.2\times 0.2)$

Similarly, for the second term in the denominator:

$0.08\times 0.08\times 0.08={(2\times 0.04)}^3=2^3\times (0.04\times 0.04\times 0.04)=8\times (0.04\times 0.04\times 0.04)$

Step 4: Factor out the common multiplier

Factoring out 8 from the entire denominator gives:

$(0.4\times 0.4\times 0.4+0.08\times 0.08\times 0.08)=8\times (0.2\times 0.2\times 0.2+0.04\times 0.04\times 0.04)$

Step 5: Substitute and simplify

Now substitute this expression back into the denominator of the main fraction:

$\frac{(0.2\times 0.2\times 0.2+0.04\times 0.04\times 0.04)}{8\times (0.2\times 0.2\times 0.2+0.04\times 0.04\times 0.04)}$

Canceling out the identical term present in both the numerator and denominator:

$\frac{1}{8}=0.125$

Therefore, the value of the given expression is 0.125.