Problem 78 - Elementary Algebra

Question

\(7.9 \times 10^{4} \mathrm{~km}+6.8 \times 10^{4} \mathrm{~km}\)

Step-by-Step Solution

Answer

14.7 \times 10^{4} \;km

Step 1: Identify the Given Values

The given values to be added are: -\tFirst value: \(7.9 \times 10^{4} \;km\)-\tSecond value: \(6.8 \times 10^{4} \;km\)

Step 2: Ensure Same Power of Ten

Both values are already expressed with the same power of ten, \(10^{4}\). Hence, no adjustments are necessary.

Step 3: Add the Base Values

Add the base values (the coefficients before the power of ten):\(7.9 + 6.8 = 14.7\)

Step 4: Combine with the Power of Ten

Combine the result with the common power of ten:\(14.7 \times 10^{4} \;km\)

Key Concepts

powers of ten

Understanding powers of ten is essential when working with scientific notation. A power of ten indicates how many times you need to multiply or divide by 10. For example, in the expression \(10^{4}\), the 4 is the power, signifying that the number 10 is multiplied by itself 4 times: \(10 \times 10 \times 10 \times 10 = 10,000\). Concepts like these help simplify and manage very large or very small numbers with ease. When adding or subtracting numbers in scientific notation, it is crucial to ensure they have the same power of ten. This allows for the easy combination of their coefficients.

addition of coefficients

In scientific notation, the numbers in front of the powers of ten are called coefficients. To add numbers in scientific notation effectively, we first look at these coefficients. For instance, in the problem \(7.9 \times 10^{4} + 6.8 \times 10^{4}\), the coefficients are 7.9 and 6.8. Since these numbers already have the same power of ten, we can directly add the coefficients together. Adding 7.9 and 6.8 gives us 14.7. This is a straightforward step but crucial for correctly handling scientific notation operations.

combining like terms

Combining like terms is a fundamental concept in algebra and scientific notation. Like terms have the same variable raised to the same power. In scientific notation, this means having the same power of ten. For example, in \(7.9 \times 10^{4} + 6.8 \times 10^{4}\), both terms have \(10^{4}\) as their power of ten, making them 'like terms'. Thus, we combine by adding their coefficients: \(7.9 + 6.8 = 14.7\). The final step is to pair this sum with the common power of ten, resulting in \(14.7 \times 10^{4}\). This principle makes the addition of numbers in scientific notation efficient and manageable.