Dividing numbers in scientific notation with exponents

Key concept: This problem measures whether you can divide scientific notation correctly by handling the decimal coefficients and powers of ten separately. That is a core skill for chemistry, physics, and algebra calculations.

Step by step

Key idea

When dividing numbers in scientific notation, do two things:

1. Divide the decimal coefficients.
2. Subtract the exponents on the powers of 10.

The general rule is:

$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

For the given expression:

$\frac{2.30 \times 10^5}{1.15 \times 10^{10}}$

Step-by-step method

First divide the coefficients:

$\frac{2.30}{1.15} = 2.00$

Then subtract the exponents:

$10^5 \div 10^{10} = 10^{5-10} = 10^{-5}$

Combine the results:

$\frac{2.30 \times 10^5}{1.15 \times 10^{10}} = 2.00 \times 10^{-5}$

Why the exponent rule matters

When dividing powers with the same base, you subtract exponents, not add them. This is one of the most tested properties in scientific notation. The coefficient and the power of ten must both be simplified before you choose the final answer.

Common mistakes

A common error is to divide the coefficients correctly but then add the exponents, which would give the wrong power of ten. Another error is to ignore the decimal places and estimate loosely. In scientific notation problems, the coefficient must usually be written so it is between 1 and 10.

Final check

The correct simplification is:

$2.00 \times 10^{-5}$

That matches the standard form for scientific notation and the correct exponent subtraction rule.

Pitfall alert

Students often focus only on the numbers and forget the exponent rule, which is why they may choose an answer like $2.00$ or even $2.00 \times 10^2$. That usually happens when they divide 2.30 by 1.15 and stop before handling the powers of ten. Another common issue is losing track of the negative exponent after subtraction. If the denominator has the larger power of ten, the final exponent will be negative. Always separate the coefficient work from the exponent work, then recombine them in standard scientific notation.

Try different conditions

If the expression were $\frac{4.2 \times 10^8}{2.1 \times 10^3}$, the coefficient quotient would be 2 and the exponent difference would be $10^{8-3}=10^5$, so the result would be $2 \times 10^5$. If the denominator were larger, such as $\frac{6.0 \times 10^4}{3.0 \times 10^9}$, the result would be $2 \times 10^{-5}$. These variants show the same method: divide the leading numbers and subtract the powers of ten.

Further reading

scientific notation, quotient rule, powers of ten