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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 \( 2.5 \times 10^3 \).

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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.

a) Work out (4 × 10^4) × (6 × 10^7) and give your answer in standard form. b) Work out (5 × 10^6) ÷ (2 × 10^3) and give your answer in standard form.

This page covers the solution to multiplication and division problems involving powers of ten. The problems include (4 × 10^4) × (6 × 10^7) and (5 × 10^6) ÷ (2 × 10^3), demonstrating how to handle scientific notation and simplify the results into standard form. Suitable for Grades 7-9.

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