To simplify the expression [tex]\( 31.5 \cdot 10.51 \)[/tex] using the rules of significant figures, let's go through the steps in detail:
1. Identify the significant figures in each value:
- The number [tex]\( 31.5 \)[/tex] has 3 significant figures.
- The number [tex]\( 10.51 \)[/tex] has 4 significant figures.
2. Perform the multiplication:
- Multiply [tex]\( 31.5 \)[/tex] by [tex]\( 10.51 \)[/tex]:
[tex]\[
31.5 \times 10.51 = 331.065
\][/tex]
3. Determine the number of significant figures in the result:
- When multiplying or dividing, the result should have the same number of significant figures as the value with the fewest significant figures.
- Here, the value with the fewest significant figures is [tex]\( 31.5 \)[/tex], which has 3 significant figures.
4. Round the result to the correct number of significant figures:
- The result [tex]\( 331.065 \)[/tex] must be rounded to 3 significant figures.
- To round [tex]\( 331.065 \)[/tex] to 3 significant figures, observe the first three significant figures: [tex]\( 3, 3, \)[/tex] and [tex]\( 1 \)[/tex]. The digit following these three figures is a [tex]\( 0 \)[/tex], which does not affect the rounding.
- Hence, [tex]\( 331.065 \)[/tex] rounded to 3 significant figures is [tex]\( 331.06 \)[/tex].
So the simplified expression using the rules of significant figures is:
[tex]\[
31.5 \cdot 10.51 = 331.06
\][/tex]
Therefore, the simplified result is [tex]\( 331.06 \)[/tex].
