Problem 48
Question
\(7 \times 10^{6}+2 \times 10^{5}\)
Step-by-Step Solution
Verified Answer
7200000
1Step 1 - Understand the Terms
The given expression is a sum of two terms involving numbers raised to powers of 10. Specifically, the terms are: \(7 \times 10^6\) and \(2 \times 10^5\).
2Step 2 - Simplify Each Term
First, calculate each term separately. For the term \(7 \times 10^6\): \[7 \times 10^6 = 7000000.\] For the term \(2 \times 10^5\): \[2 \times 10^5 = 200000.\]
3Step 3 - Add the Results
Now add the two results obtained from the previous step. \[7000000 + 200000 = 7200000.\]
Key Concepts
ExponentiationAddition of Large NumbersSimplification of Expressions
Exponentiation
Exponentiation is a mathematical operation that involves raising a number, called the base, to a specified power, called the exponent. In this exercise, we encounter numbers like \(10^6\) and \(10^5\). These expressions mean:
Understanding exponentiation is crucial for working with scientific notation, which allows us to handle very large numbers more easily.
- \(10^6\) is 10 raised to the power of 6, or 10 multiplied by itself 6 times: \[ 10^6 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000. \]
- \(10^5\) is 10 raised to the power of 5, or 10 multiplied by itself 5 times: \[ 10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000. \]
Understanding exponentiation is crucial for working with scientific notation, which allows us to handle very large numbers more easily.
Addition of Large Numbers
Adding large numbers can be daunting, but it becomes manageable by breaking down the process. In this exercise, we've simplified our terms using exponentiation. After calculating each term separately, we get:
Now, we simply add these large numbers:
\[ 7,000,000 + 200,000 = 7,200,000. \]
It helps to line up the numbers according to their place values and add them column by column, just like you would with smaller numbers.
- \(7 \times 10^6 = 7,000,000\)
- \(2 \times 10^5 = 200,000\)
Now, we simply add these large numbers:
\[ 7,000,000 + 200,000 = 7,200,000. \]
It helps to line up the numbers according to their place values and add them column by column, just like you would with smaller numbers.
Simplification of Expressions
Simplifying expressions in scientific notation involves a few clear steps. First, each term is converted from its scientific notation form to standard form. Then, we perform the necessary arithmetic operations. Let's look at our example:
Given: \[7 \times 10^6 + 2 \times 10^5 \]
Converting to standard form:
Adding the results:
\[ 7,000,000 + 200,000 = 7,200,000 \]
The final simplified expression in standard form is 7,200,000. Understanding and following these steps makes it easier to simplify complex expressions accurately.
Given: \[7 \times 10^6 + 2 \times 10^5 \]
Converting to standard form:
- \(7 \times 10^6 = 7,000,000\)
- \(2 \times 10^5 = 200,000\)
Adding the results:
\[ 7,000,000 + 200,000 = 7,200,000 \]
The final simplified expression in standard form is 7,200,000. Understanding and following these steps makes it easier to simplify complex expressions accurately.