(a) Write 76 000 000 in standard form. (1) (b) Write 8.3 × 10–4 as an ordinary number. (1) (c) Calculate (3 × 10147) ÷ (6 × 10122) Give your answer in standard form. (2) (d) Write 1.6 × 1041 as a product of its prime factors. Show your working clearly. 

Let's tackle each part of the problem step by step.

### (a) Write 76,000,000 in standard form. (1 mark)

To write \(76,000,000\) in standard form:

1. Place the decimal point after the first non-zero digit: \(7.6\).
2. Count the number of places the decimal point has moved from its original position (after the last zero) to after the first digit. Here, it moved 7 places.

So, the standard form is:
\[
7.6 \times 10^7
\]

### (b) Write \(8.3 \times 10^{-4}\) as an ordinary number. (1 mark)

To convert \(8.3 \times 10^{-4}\) to an ordinary number:

1. The negative exponent \(-4\) means we move the decimal point 4 places to the left.
2. Starting from \(8.3\), move the decimal 4 places left: \(0.00083\).

So, the ordinary number is:
\[
0.00083
\]

### (c) Calculate \(\frac{3 \times 10^{147}}{6 \times 10^{122}}\) and give your answer in standard form. (2 marks)

To calculate \(\frac{3 \times 10^{147}}{6 \times 10^{122}}\):

1. Divide the coefficients: \(\frac{3}{6} = 0.5\).
2. Subtract the exponents of 10: \(147 - 122 = 25\).

So, the result is:
\[
0.5 \times 10^{25}
\]

To express this in standard form, rewrite \(0.5\) as \(5 \times 10^{-1}\):
\[
0.5 \times 10^{25} = 5 \times 10^{-1} \times 10^{25} = 5 \times 10^{24}
\]

### (d) Write \(1.6 \times 10^{41}\) as a product of its prime factors. Show your working clearly.

1. The number \(1.6\) is the coefficient, and \(10^{41}\) represents \(10\) raised to the power of \(41\).
2. The prime factorization of \(10\) is \(2 \times 5\).
3. Therefore, \(10^{41} = (2 \times 5)^{41} = 2^{41} \times 5^{41}\).

Now, express \(1.6\) as a product of prime factors:
1. \(1.6\) can be written as \(16 \times 10^{-1} = 2^4 \times 10^{-1}\).
2. Using \(10^{-1} = 2^{-1} \times 5^{-1}\), we get \(1.6 = 2^4 \times 2^{-1} \times 5^{-1} = 2^3 \times 5^{-1}\).

So:
\[
1.6 \times 10^{41} = (2^3 \times 5^{-1}) \times (2^{41} \times 5^{41}) = 2^{44} \times 5^{40}
\]

Would you like a more detailed explanation, or do you have any questions?

Here are some related problems you might find interesting:
1. Write \(0.00045\) in standard form.
2. Convert \(7.2 \times 10^3\) into an ordinary number.
3. Simplify \(\frac{4 \times 10^{10}}{2 \times 10^6}\) and express your answer in standard form.
4. Express \(3.4 \times 10^{-2}\) as a fraction.
5. What is the prime factorization of \(3.2 \times 10^{20}\)?

**Tip:** When working with powers of 10, remember that multiplying or dividing the coefficients separately from the powers can simplify the process.

Explore mathematical problem-solving techniques including converting numbers to standard form, scientific notation, and prime factorization. Learn step-by-step solutions to problems like writing numbers in standard form, calculating with scientific notation, and finding prime factors. Suitable for high school students.

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