If p = (12.34 × 10¹⁰) – (5.67 × 10⁹), then p is expressed in standard form as:

Concept: A number is in standard form (scientific notation) when it is written as a × 10ⁿ with 1 ≤ a < 10 and n an integer (it may be negative). Two numbers written this way can be added or subtracted only after both are rewritten with the same power of ten; the mantissas are then combined over that common power, and the result is renormalised so that its mantissa again lies between 1 and 10.

Application: Apply that rule to p = (12.34 × 10¹⁰) − (5.67 × 10⁹).

1. Match the powers of ten. Moving the decimal point one place to the right multiplies the mantissa by 10, so the exponent must drop by 1: 12.34 × 10¹⁰ = 123.4 × 10⁹.

2. The second term already carries the power 10⁹, so it stays as it is: 5.67 × 10⁹.

3. Subtract the mantissas over the common power: p = (123.4 − 5.67) × 10⁹ = 117.73 × 10⁹.

4. Renormalise. 117.73 does not lie between 1 and 10, and 117.73 = 1.1773 × 10², so p = 1.1773 × 10² × 10⁹ = 1.1773 × 10¹¹.

Cross-check: Expand both terms fully: 12.34 × 10¹⁰ = 123,400,000,000 and 5.67 × 10⁹ = 5,670,000,000. Subtracting gives 123,400,000,000 − 5,670,000,000 = 117,730,000,000. Counting 11 places from the leading digit rewrites that as 1.1773 × 10¹¹, whose mantissa 1.1773 lies between 1 and 10, so the form is standard.

Where near-misses come from: The mantissa digits and the exponent are two separate things to get right. Losing a digit while writing 117.73 leaves the mantissa 1.773 instead of 1.1773, and shifting the decimal point only one place instead of two leaves the exponent at 10 instead of 11. Either slip changes the value by a factor of ten or more, even though the result still looks like standard form.

Result: p = 1.1773 × 10¹¹.