Let's work through the expression step by step.

The given expression is:

$\frac{3.56 \times 10^{-5} \times 5.87 \times 10^{12}}{4.03 \times 10^{3}}$

Step 1: Simplify the powers of 10

First, handle the powers of 10 separately.

$(10^{-5} \times 10^{12}) = 10^{-5 + 12} = 10^{7}$

So, the expression becomes:

$\frac{3.56 \times 5.87 \times 10^{7}}{4.03 \times 10^{3}}$

Step 2: Simplify the numerical part

Now, multiply the numbers in the numerator and denominator:

$3.56 \times 5.87 = 20.91$

The expression is now:

$\frac{20.91 \times 10^{7}}{4.03 \times 10^{3}}$

Step 3: Simplify the powers of 10

Now divide the powers of 10:

$\frac{10^{7}}{10^{3}} = 10^{7 - 3} = 10^{4}$

So, the expression is now:

$\frac{20.91}{4.03} \times 10^{4}$

Step 4: Simplify the division

Now divide $20.91$ by $4.03$:

$\frac{20.91}{4.03} \approx 5.19$

So the expression becomes:

$5.19 \times 10^{4}$

Step 5: Final Answer in Standard Form

To 3 significant figures, the answer is:

$5.19 \times 10^{4}$

Would you like a more detailed breakdown or have any other questions on this?

Related questions:

1. How do we handle powers of 10 when multiplying and dividing scientific notation?
2. How do you round numbers to a specific number of significant figures?
3. What are the rules for simplifying powers of 10 in scientific notation?
4. Can you convert the number 5.19 × 10⁴ to standard decimal form?
5. How does scientific notation help when working with very large or small numbers?

Tip:

When working with scientific notation, always treat the base (like 10) separately from the coefficients for easier calculation.