8. Choose the appropriate sign for each of the following expressions in the table. 1.98 x 10^5 ? 8.91 x 10^5, 2.14 x 10^5 ? 2.4 x 10^3, 8.341 x 10^-6 ? 2.15 x 10^-2. The options for ? are <, >, or =.
1.98 x 10^5 < 8.91 x 10^5
2.14 x 10^5 > 2.4 x 10^3
8.341 x 10^-6 < 2.15 x 10^-2
To compare the expressions, we can compare the coefficients first, and then compare the exponents.
1.98 x 10^5 ? 8.91 x 10^5:
The coefficients are 1.98 and 8.91. Since 8.91 is greater than 1.98, the appropriate sign is >.
2.14 x 10^5 ? 2.4 x 10^3:
The coefficients are 2.14 and 2.4. Since 2.14 is greater than 2.4, the appropriate sign is >.
8.341 x 10^-6 ? 2.15 x 10^-2:
The coefficients are 8.341 and 2.15. Since 8.341 is much smaller than 2.15, the appropriate sign is <.
So, the appropriate signs for the expressions are:
1.98 x 10^5 > 8.91 x 10^5
2.14 x 10^5 > 2.4 x 10^3
8.341 x 10^-6 < 2.15 x 10^-2
To determine the appropriate sign for each of the given expressions, we need to compare the values represented in scientific notation.
1.98 x 10^5 ? 8.91 x 10^5
To compare these two expressions, we only need to consider the coefficients (the numbers in front of the 10). In this case, 1.98 is smaller than 8.91. Since both expressions have the same exponent (10^5), we can conclude that 1.98 x 10^5 is less than 8.91 x 10^5. Therefore, the appropriate sign for this expression is < (less than).
2.14 x 10^5 ? 2.4 x 10^3
Again, we only need to compare the coefficients. In this case, 2.14 is greater than 2.4. Since both expressions have the same exponent (10^5 and 10^3), we cannot directly compare them. To make the exponents of the two expressions equal, we need to adjust the 10^3 expression.
Since 10^5 is equal to 10^3 x 10^2, we can convert 2.4 x 10^3 into scientific notation with an exponent of 10^5. Thus, 2.4 x 10^3 becomes 2.4 x 10^3 x 10^2 = 2.4 x 10^5. Now we can compare the coefficients directly: 2.14 is still greater than 2.4, so 2.14 x 10^5 is greater than 2.4 x 10^3. Therefore, the appropriate sign for this expression is > (greater than).
8.341 x 10^-6 ? 2.15 x 10^-2
Similar to the previous comparisons, we compare the coefficients. In this case, 8.341 is smaller than 2.15. However, the exponents are different: -6 and -2. To compare them directly, we need to adjust the exponents.
Since a negative exponent represents a fraction, and going from a smaller negative exponent to a larger negative exponent means moving to a smaller fraction, we can conclude that 10^-6 is greater than 10^-2. Therefore, we can indirectly say that 8.341 x 10^-6 is greater than 2.15 x 10^-2.
To summarize:
1.98 x 10^5 < 8.91 x 10^5
2.14 x 10^5 > 2.4 x 10^3
8.341 x 10^-6 > 2.15 x 10^-2
The appropriate signs for the given expressions are <, >, and >, respectively.