For numbers greater than or equal to 10, follow these steps:

1. Move the decimal point to the left until you get a number between 1 and 10.
2. Count how many places you moved the decimal.
3. The exponent n is positive and equals the number of places moved.

Example 1: Write 230,000,000 in standard form.

Solution

• Original number: 230,000,000
• Move decimal 8 places to the left: 2.30000000 = 2.3
• The exponent is 8 (positive because we moved left)

$230,000,000 = 2.3 \times 10^8$

Example 2: Write 245 in standard form.

Solution

• Move decimal 2 places to the left: 2.45
• Exponent is 2

$245 = 2.45 \times 10^2$

For numbers between 0 and 1, follow these steps:

1. Move the decimal point to the right until you get a number between 1 and 10.
2. Count how many places you moved the decimal.
3. The exponent n is negative and equals the number of places moved.

Example 3: Write 0.00045 in standard form.

Solution

• Move decimal 4 places to the right: 4.5
• The exponent is -4 (negative because we moved right)

$0.00045 = 4.5 \times 10^{-4}$

Example 4: Write 0.000008 in standard form.

Solution

• Move decimal 6 places to the right: 8
• Exponent is -6

$0.000008 = 8 \times 10^{-6}$

To convert a number from standard form to ordinary decimal form:

• If the exponent is positive, move the decimal point to the right.
• If the exponent is negative, move the decimal point to the left.

Example 5: Write $4.8 \times 10^3$ in ordinary form.

Solution

Move decimal 3 places to the right:

$4.8 \times 10^3 = 4.8 \times 1000 = 4800$

Example 6: Write $3.5 \times 10^{-6}$ in ordinary form.

Solution

Move decimal 6 places to the left:

$3.5 \times 10^{-6} = 3.5 \times 0.000001 = 0.0000035$

When multiplying numbers in standard form, add the exponents:

$(A \times 10^n)(B \times 10^m) = (A \times B) \times 10^{n+m}$

Example 7: Simplify $(7 \times 10^2)(8 \times 10^4)$

Solution

$(7 \times 10^2)(8 \times 10^4) = (7 \times 8) \times 10^{2+4} = 56 \times 10^6$

Now convert 56 to standard form:

$56 \times 10^6 = 5.6 \times 10 \times 10^6 = 5.6 \times 10^7$

When dividing, subtract the exponents:

$\frac{A \times 10^n}{B \times 10^m} = \frac{A}{B} \times 10^{n-m}$

Example 8: Evaluate $\frac{4 \times 10^8}{5 \times 10^5}$

Solution

$\frac{4 \times 10^8}{5 \times 10^5} = \frac{4}{5} \times 10^{8-5} = 0.8 \times 10^3 = 8 \times 10^2$

• The coefficient must be at least 1 but less than 10 (1 ≤ A < 10)
• The exponent can be positive (for large numbers) or negative (for small numbers)
• A positive exponent means the number is greater than or equal to 10
• A negative exponent means the number is between 0 and 1
• When multiplying, add the exponents; when dividing, subtract them

In Tanzania, standard form is used in many everyday situations. For example, when the government announces the national budget of about 40 trillion Tanzanian shillings, it can be written as $4 \times 10^{13}$ TZS — this makes it easier to read and compare with budgets from other years. Similarly, a small business owner calculating profits or a farmer tracking crop yields may encounter very large or very small numbers that are best expressed in standard form.