Express the value or answer in powers of 10 to the correct number of significant figures:
4586.49
2450
Square of 199.99
Sum of (32.087 + 1.56 + 206.44) divided by 12.3
The correct answer and explanation is :
Let’s break down each of the calculations step by step, focusing on expressing the answers in powers of 10 to the correct number of significant figures.
1. 4586.49
To express 4586.49 in powers of 10, we first note the number of significant figures. The number 4586.49 has 6 significant figures. When expressed in scientific notation:
- 4586.49 = ( 4.58649 \times 10^3 )
So the value in powers of 10 is ( 4.58649 \times 10^3 ), maintaining the 6 significant figures.
2. 2450
For 2450, we need to determine the number of significant figures. Since no decimal point is present, this number could have either 3 or 4 significant figures depending on context. But, typically in scientific notation:
- If we assume 3 significant figures, the number would be expressed as:
2450 = ( 2.45 \times 10^3 ) - If we assume 4 significant figures, it would be:
2450 = ( 2.450 \times 10^3 )
For this answer, we’ll assume the more common case of 3 significant figures and express it as ( 2.45 \times 10^3 ).
3. Square of 199.99
The square of 199.99 is calculated as:
[
199.99^2 = 39996.0001
]
Now, let’s focus on the significant figures in the input. 199.99 has 5 significant figures. Since we are squaring it, the result should also be expressed with 5 significant figures:
- 39996.0001 ≈ ( 4.0000 \times 10^4 ) (rounded to 5 significant figures).
4. Sum of (32.087 + 1.56 + 206.44) divided by 12.3
First, we calculate the sum of the values:
[
32.087 + 1.56 + 206.44 = 240.087
]
Now, divide this sum by 12.3:
[
\frac{240.087}{12.3} = 19.5
]
When considering significant figures, 32.087 has 5 significant figures, 1.56 has 3 significant figures, 206.44 has 5 significant figures, and 12.3 has 3 significant figures. The result should be reported to 2 significant figures because division is limited by the lowest number of significant figures in the operands.
So, the final result in powers of 10 is:
[
19.5 \approx 1.95 \times 10^1
]
Conclusion
The answers in powers of 10, to the correct number of significant figures, are:
- 4586.49 = ( 4.58649 \times 10^3 )
- 2450 = ( 2.45 \times 10^3 ) (assuming 3 significant figures)
- Square of 199.99 = ( 4.0000 \times 10^4 )
- Sum of (32.087 + 1.56 + 206.44) divided by 12.3 = ( 1.95 \times 10^1 )
Explanation
In scientific notation, it is essential to express values with the correct number of significant figures based on the precision of the given numbers. Significant figures indicate the precision of a measurement. When performing mathematical operations like addition, subtraction, multiplication, or division, the result should be rounded according to the number of significant figures in the input values. This ensures that the result reflects the precision of the measurements provided. For example, when summing 32.087, 1.56, and 206.44, the least precise number (1.56, with 3 significant figures) dictates that the final answer should be rounded to 2 significant figures after division, as shown in the final answer of ( 1.95 \times 10^1 ). This maintains consistency and accuracy in scientific reporting.