Which is the best estimate for \frac{(8.9 \times 10^8)}{(3.3 \times 10^4)} written in scientific notation?
The Correct Answer and Explanation is:1
Correct Answer: 3 x 10⁴
Explanation:
To find the best estimate for the expression \frac{(8.9 \times 10^8)}{(3.3 \times 10^4)}, we can round the coefficients to the nearest whole numbers to simplify the calculation. This approach allows us to quickly approximate the answer.
The original expression is:
8.9×1083.3×1043.3×1048.9×108Step 1: Estimate the coefficients
First, round the decimal numbers (the coefficients) to the nearest integer to make the division easier.
- 8.9 is very close to 9.
- 3.3 is very close to 3.
Now, substitute these rounded numbers back into the expression:
≈9×1083×104≈3×1049×108Step 2: Divide the coefficients and the powers of 10 separately
When dividing numbers in scientific notation, the process is broken into two parts: dividing the coefficients and dividing the powers of 10.
- Divide the coefficients:
9÷3=39÷3=3 - Divide the powers of 10:
To divide powers with the same base, we use the quotient rule of exponents, which states thatam÷an=am−nam÷an=am−n. In this case, we subtract the exponent in the denominator from the exponent in the numerator.108÷104=10(8−4)=104108÷104=10(8−4)=104
Step 3: Combine the results
Finally, combine the results from the two parts to get the estimated answer in scientific notation.
The new coefficient is 3, and the new power of 10 is 10⁴.
Therefore, the best estimate is:
3×1043×104This result is already in proper scientific notation, as the coefficient (3) is a number between 1 and 10. Comparing this estimate with the given options, we find that 3 x 10⁴ is one of the choices.
