Work Out \left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right ].Give Your Answer In Standard Index Form.

Understanding Standard Index Form

Standard index form, also known as scientific notation, is a way of expressing very large or very small numbers in a more manageable and concise form. It consists of a number between 1 and 10 multiplied by a power of 10. This format is widely used in mathematics, physics, and engineering to simplify complex calculations and make them more efficient.

The Importance of Standard Index Form

Standard index form is essential in mathematics and science because it allows us to express numbers in a more compact and readable format. This format is particularly useful when dealing with very large or very small numbers, such as those encountered in astronomy, chemistry, and physics. By expressing numbers in standard index form, we can perform calculations more easily and accurately.

Converting Between Standard Index Form and Decimal Form

To convert a number from standard index form to decimal form, we need to multiply the number by the corresponding power of 10. For example, to convert the number $3.4 \times 10^5$ to decimal form, we multiply 3.4 by $10^5$, which gives us $3.4 \times 10^5 = 340,000$.

Converting Between Standard Index Form and Decimal Form: Example

Let's consider the example of converting the number $2.5 \times 10^3$ to decimal form. To do this, we multiply 2.5 by $10^3$, which gives us $2.5 \times 10^3 = 2,500$.

Adding and Subtracting Numbers in Standard Index Form

When adding or subtracting numbers in standard index form, we need to ensure that the powers of 10 are the same. If the powers of 10 are the same, we can add or subtract the numbers directly. If the powers of 10 are different, we need to adjust the numbers by multiplying or dividing them by the corresponding power of 10.

Adding and Subtracting Numbers in Standard Index Form: Example

Let's consider the example of adding the numbers $3.2 \times 10^4$ and $2.1 \times 10^4$. Since the powers of 10 are the same, we can add the numbers directly: $3.2 \times 10^4 + 2.1 \times 10^4 = 5.3 \times 10^4$.

Multiplying and Dividing Numbers in Standard Index Form

When multiplying or dividing numbers in standard index form, we need to multiply or divide the numbers directly and then add or subtract the powers of 10. For example, to multiply the numbers $2.5 \times 10^3$ and $3.4 \times 10^2$, we multiply 2.5 by 3.4 and add the powers of 10: $(2.5 \times 10^3) \times (3.4 \times 10^2) = 8.5 \times 10^5$.

Multiplying and Dividing Numbers in Standard Index Form: Example

Let's consider the example of dividing the numbers $4.2 \times 10^5$ by $2.1 \times 10^3$. To do this, we divide 4.2 by 2.1 and subtract the powers of 10: $(4.2 \times 10^5) \div (2.1 \times 10^3) = 2 \times 10^2$.

Work Out $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$

To work out the expression $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$, we need to ensure that the powers of 10 are the same. Since the powers of 10 are different, we need to adjust the numbers by multiplying or dividing them by the corresponding power of 10.

Work Out $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$: Solution

To adjust the numbers, we can multiply the first number by $10^1$ and the second number by $10^{-1}$. This gives us $\left(5.2 \times 10^7\right) \times 10^1 = 5.2 \times 10^8$ and $\left(1.4 \times 10^8\right) \times 10^{-1} = 1.4 \times 10^7$.

Work Out $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$: Solution

Now that the powers of 10 are the same, we can add the numbers directly: $5.2 \times 10^8 + 1.4 \times 10^7 = 5.2 \times 10^8 + 0.14 \times 10^8 = 5.34 \times 10^8$.

Conclusion

Standard index form is a powerful tool for simplifying large numbers and making calculations more efficient. By understanding how to convert between standard index form and decimal form, add and subtract numbers in standard index form, and multiply and divide numbers in standard index form, we can perform complex calculations with ease. In this article, we have worked out the expression $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$ and obtained the solution $5.34 \times 10^8$.

Understanding Standard Index Form

Standard index form, also known as scientific notation, is a way of expressing very large or very small numbers in a more manageable and concise form. It consists of a number between 1 and 10 multiplied by a power of 10. This format is widely used in mathematics, physics, and engineering to simplify complex calculations and make them more efficient.

Q&A: Standard Index Form

Q: What is standard index form?

A: Standard index form is a way of expressing very large or very small numbers in a more manageable and concise form. It consists of a number between 1 and 10 multiplied by a power of 10.

Q: Why is standard index form important?

A: Standard index form is essential in mathematics and science because it allows us to express numbers in a more compact and readable format. This format is particularly useful when dealing with very large or very small numbers, such as those encountered in astronomy, chemistry, and physics.

Q: How do I convert a number from standard index form to decimal form?

A: To convert a number from standard index form to decimal form, you need to multiply the number by the corresponding power of 10. For example, to convert the number $3.4 \times 10^5$ to decimal form, you multiply 3.4 by $10^5$, which gives you $3.4 \times 10^5 = 340,000$.

Q: How do I add and subtract numbers in standard index form?

A: When adding or subtracting numbers in standard index form, you need to ensure that the powers of 10 are the same. If the powers of 10 are the same, you can add or subtract the numbers directly. If the powers of 10 are different, you need to adjust the numbers by multiplying or dividing them by the corresponding power of 10.

Q: How do I multiply and divide numbers in standard index form?

A: When multiplying or dividing numbers in standard index form, you need to multiply or divide the numbers directly and then add or subtract the powers of 10. For example, to multiply the numbers $2.5 \times 10^3$ and $3.4 \times 10^2$, you multiply 2.5 by 3.4 and add the powers of 10: $(2.5 \times 10^3) \times (3.4 \times 10^2) = 8.5 \times 10^5$.

Q: What is the solution to the expression $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$?

A: To work out the expression $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$, we need to ensure that the powers of 10 are the same. Since the powers of 10 are different, we need to adjust the numbers by multiplying or dividing them by the corresponding power of 10. The solution is $5.34 \times 10^8$.

Common Mistakes to Avoid

Mistake 1: Not ensuring the powers of 10 are the same when adding or subtracting numbers

When adding or subtracting numbers in standard index form, it is essential to ensure that the powers of 10 are the same. If the powers of 10 are different, you need to adjust the numbers by multiplying or dividing them by the corresponding power of 10.

Mistake 2: Not multiplying or dividing the numbers correctly when multiplying or dividing numbers

When multiplying or dividing numbers in standard index form, you need to multiply or divide the numbers directly and then add or subtract the powers of 10. For example, to multiply the numbers $2.5 \times 10^3$ and $3.4 \times 10^2$, you multiply 2.5 by 3.4 and add the powers of 10: $(2.5 \times 10^3) \times (3.4 \times 10^2) = 8.5 \times 10^5$.

Conclusion

Standard index form is a powerful tool for simplifying large numbers and making calculations more efficient. By understanding how to convert between standard index form and decimal form, add and subtract numbers in standard index form, and multiply and divide numbers in standard index form, you can perform complex calculations with ease. In this article, we have worked out the expression $\left(5.2 \times 10^7\right) + \left(1.4 \times 10^8\right)$ and obtained the solution $5.34 \times 10^8$.