Let's work through the questions.

Part a:

You are asked to calculate $(4 \times 10^4) \times (6 \times 10^7)$.

Step 1: Multiply the numbers (coefficients)

$4 \times 6 = 24$

Step 2: Multiply the powers of 10

When you multiply powers of 10, you add the exponents: $10^4 \times 10^7 = 10^{4+7} = 10^{11}$

Step 3: Combine the results

So, the result is: $24 \times 10^{11}$

However, in standard form, the number must be written with a single digit before the decimal. So, we adjust the coefficient: $24 = 2.4 \times 10^1$

Now combine this with the exponent: $2.4 \times 10^{1+11} = 2.4 \times 10^{12}$

Thus, the answer for a) is $2.4 \times 10^{12}$.

Part b:

You are asked to calculate $(5 \times 10^6) \div (2 \times 10^3)$.

Step 1: Divide the numbers (coefficients)

$5 \div 2 = 2.5$

Step 2: Divide the powers of 10

When you divide powers of 10, you subtract the exponents: $10^6 \div 10^3 = 10^{6-3} = 10^3$

Step 3: Combine the results

So, the result is: $2.5 \times 10^3$

Thus, the answer for b) is $2.5 \times 10^3$.

Do you have any questions about these calculations or want further details? Here are some related questions you could explore:

1. What would the result be if you multiplied $(2 \times 10^5) \times (3 \times 10^6)$?
2. How do you handle division when both numbers are in standard form?
3. What is the general rule for adding exponents when multiplying powers of ten?
4. How do you convert any large number into standard form?
5. What are the key differences between scientific and standard notation?

Tip: When working with powers of ten, always ensure that the coefficient in standard form is between 1 and 10 by adjusting the exponent as necessary.