Divide:$\frac{4 \times 10^2}{5 \times 10^4}$A. $8 \times 10^{\circ}$B. $6 \times 10^0$C. $8 \times 10^6$D. $8 \times 10^{-3}$

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Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10, multiplied by a power of 10. For example, the number 400 can be written in scientific notation as 4 × 10^2. Similarly, the number 0.0004 can be written as 4 × 10^-4.

Dividing Numbers in Scientific Notation

When dividing numbers in scientific notation, we need to follow a specific procedure. The procedure involves dividing the numbers and then subtracting the exponents of the powers of 10.

Step 1: Divide the Numbers

To divide numbers in scientific notation, we first divide the numbers themselves. For example, if we want to divide 4 × 10^2 by 5 × 10^4, we would first divide 4 by 5, which gives us 0.8.

Step 2: Subtract the Exponents

Next, we subtract the exponents of the powers of 10. In this case, we have 10^2 in the numerator and 10^4 in the denominator. To subtract the exponents, we subtract 4 from 2, which gives us -2.

Step 3: Write the Result

Now that we have divided the numbers and subtracted the exponents, we can write the result. The result is 0.8 × 10^-2.

Simplifying the Result

To simplify the result, we can express it in a more conventional form. We can rewrite 0.8 × 10^-2 as 8 × 10^-3.

Conclusion

In conclusion, dividing numbers in scientific notation involves dividing the numbers and then subtracting the exponents of the powers of 10. By following this procedure, we can simplify complex numbers and express them in a more manageable form.

Example Questions

A. 8×10∘8 \times 10^{\circ}

To solve this problem, we need to understand that the exponent is missing. In this case, we can assume that the exponent is 0, since any number raised to the power of 0 is equal to 1.

B. 6×1006 \times 10^0

This problem is similar to the previous one. Since the exponent is 0, we can assume that the number is equal to 1.

C. 8×1068 \times 10^6

To solve this problem, we need to understand that the exponent is 6. This means that the number is equal to 8 multiplied by 10 raised to the power of 6.

D. 8×10−38 \times 10^{-3}

This problem is similar to the previous one. Since the exponent is -3, we can assume that the number is equal to 8 multiplied by 10 raised to the power of -3.

Answer Key

  • A. 8×10∘8 \times 10^{\circ}: This problem is incomplete, and we cannot provide a final answer.
  • B. 6×1006 \times 10^0: This problem is incomplete, and we cannot provide a final answer.
  • C. 8×1068 \times 10^6: This is the correct answer.
  • D. 8×10−38 \times 10^{-3}: This is the correct answer.

Final Answer

Frequently Asked Questions

Q: What is scientific notation?

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

Q: How do I divide numbers in scientific notation?

A: To divide numbers in scientific notation, you need to follow a specific procedure. First, divide the numbers themselves. Then, subtract the exponents of the powers of 10.

Q: What if the exponents are the same?

A: If the exponents are the same, you can simply divide the numbers themselves. For example, if you want to divide 4 × 10^2 by 4 × 10^2, you would simply divide 4 by 4, which gives you 1.

Q: What if the exponents are different?

A: If the exponents are different, you need to subtract the exponents of the powers of 10. For example, if you want to divide 4 × 10^2 by 5 × 10^4, you would first divide 4 by 5, which gives you 0.8. Then, you would subtract the exponents, which gives you -2.

Q: How do I simplify the result?

A: To simplify the result, you can express it in a more conventional form. For example, if you get 0.8 × 10^-2, you can rewrite it as 8 × 10^-3.

Q: What if I get a negative exponent?

A: If you get a negative exponent, you can simply rewrite it as a positive exponent with a negative coefficient. For example, if you get 4 × 10^-3, you can rewrite it as 0.004.

Q: Can I use a calculator to divide numbers in scientific notation?

A: Yes, you can use a calculator to divide numbers in scientific notation. However, make sure to enter the numbers in the correct format, with the exponent in the correct position.

Q: What if I'm dividing a number by a decimal?

A: If you're dividing a number by a decimal, you can simply multiply the number by the reciprocal of the decimal. For example, if you want to divide 4 × 10^2 by 0.5, you can multiply 4 × 10^2 by 2, which gives you 8 × 10^2.

Q: What if I'm dividing a number by a fraction?

A: If you're dividing a number by a fraction, you can simply multiply the number by the reciprocal of the fraction. For example, if you want to divide 4 × 10^2 by 1/2, you can multiply 4 × 10^2 by 2, which gives you 8 × 10^2.

Common Mistakes

Mistake 1: Not following the correct procedure

A: Make sure to follow the correct procedure when dividing numbers in scientific notation. First, divide the numbers themselves. Then, subtract the exponents of the powers of 10.

Mistake 2: Not simplifying the result

A: Make sure to simplify the result by expressing it in a more conventional form. For example, if you get 0.8 × 10^-2, you can rewrite it as 8 × 10^-3.

Mistake 3: Not using the correct format

A: Make sure to enter the numbers in the correct format, with the exponent in the correct position. For example, if you want to divide 4 × 10^2 by 5 × 10^4, you should enter the numbers as 4 × 10^2 and 5 × 10^4.

Conclusion

Dividing numbers in scientific notation can be a challenging task, but with practice and patience, you can master it. Remember to follow the correct procedure, simplify the result, and use the correct format. With these tips and tricks, you'll be able to divide numbers in scientific notation like a pro!