Evaluate The Expression: $0.04 \cdot \left(1 \times 10^0\right)$

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Introduction

In mathematics, the concept of scientific notation is a powerful tool for representing very large or very small numbers in a more manageable form. The expression $0.04 \cdot \left(1 \times 10^0\right)$ is a simple example of how to evaluate an expression involving scientific notation. In this article, we will delve into the details of evaluating this expression and explore the underlying mathematical concepts.

Understanding Scientific Notation

Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. The general form of a number in scientific notation is $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. For example, the number 456 can be written in scientific notation as $4.56 \times 10^2$.

Evaluating the Expression

To evaluate the expression $0.04 \cdot \left(1 \times 10^0\right)$, we need to understand the properties of exponents and how to multiply numbers in scientific notation. The expression can be rewritten as $0.04 \cdot 1 \cdot 10^0$.

Multiplying Numbers in Scientific Notation

When multiplying numbers in scientific notation, we multiply the numbers and add the exponents. In this case, we have $0.04 \cdot 1 \cdot 10^0$. Since $0.04$ can be written as $4 \times 10^{-2}$, we can rewrite the expression as $(4 \times 10^{-2}) \cdot 1 \cdot 10^0$.

Applying the Rules of Exponents

Now, we can apply the rules of exponents to simplify the expression. When multiplying numbers in scientific notation, we add the exponents. In this case, we have $(4 \times 10^{-2}) \cdot 1 \cdot 10^0$. Since $1$ can be written as $10^0$, we can rewrite the expression as $(4 \times 10^{-2}) \cdot (10^0 \cdot 10^0)$.

Simplifying the Expression

Using the rule that $a^m \cdot a^n = a^{m+n}$, we can simplify the expression as $(4 \times 10^{-2}) \cdot (10^0 \cdot 10^0) = (4 \times 10^{-2}) \cdot 10^0 \cdot 10^0 = (4 \times 10^{-2}) \cdot 10^0$.

Final Evaluation

Now, we can evaluate the expression by multiplying the numbers and adding the exponents. We have $(4 \times 10^{-2}) \cdot 10^0 = 4 \times 10^{-2} \cdot 10^0 = 4 \times 10^{-2}$.

Conclusion

In conclusion, the expression $0.04 \cdot \left(1 \times 10^0\right)$ can be evaluated by rewriting it in scientific notation and applying the rules of exponents. The final answer is $4 \times 10^{-2}$, which is equivalent to $0.04$.

Frequently Asked Questions

  • Q: What is scientific notation? A: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10.
  • Q: How do I evaluate an expression involving scientific notation? A: To evaluate an expression involving scientific notation, you need to understand the properties of exponents and how to multiply numbers in scientific notation.
  • Q: What is the rule for multiplying numbers in scientific notation? A: When multiplying numbers in scientific notation, you multiply the numbers and add the exponents.

Further Reading

  • Scientific Notation: A Guide to Understanding and Evaluating Expressions
  • Exponents: A Comprehensive Guide to Understanding and Applying the Rules
  • Multiplying Numbers in Scientific Notation: A Step-by-Step Guide

References

  • [1] "Scientific Notation" by Math Open Reference
  • [2] "Exponents" by Khan Academy
  • [3] "Multiplying Numbers in Scientific Notation" by IXL

Introduction

Evaluating expressions in scientific notation can be a challenging task, especially for those who are new to the concept. In this article, we will provide a comprehensive Q&A guide to help you understand and evaluate expressions in scientific notation.

Q: What is scientific notation?

A: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. The general form of a number in scientific notation is $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer.

Q: How do I convert a number to scientific notation?

A: To convert a number to scientific notation, you need to move the decimal point to the left or right until you have a number between 1 and 10. Then, you multiply the number by 10 raised to the power of the number of places you moved the decimal point.

Q: What is the rule for multiplying numbers in scientific notation?

A: When multiplying numbers in scientific notation, you multiply the numbers and add the exponents. For example, $(3 \times 10^2) \cdot (4 \times 10^3) = 12 \times 10^{2+3} = 12 \times 10^5$.

Q: What is the rule for dividing numbers in scientific notation?

A: When dividing numbers in scientific notation, you divide the numbers and subtract the exponents. For example, $(6 \times 10^4) \div (2 \times 10^3) = 3 \times 10^{4-3} = 3 \times 10^1$.

Q: How do I evaluate an expression with multiple terms in scientific notation?

A: To evaluate an expression with multiple terms in scientific notation, you need to follow the order of operations (PEMDAS). First, evaluate any expressions inside parentheses. Then, multiply or divide the terms as needed. Finally, add or subtract the terms as needed.

Q: What is the difference between scientific notation and exponential notation?

A: Scientific notation and exponential notation are both used to represent very large or very small numbers, but they are not the same thing. Scientific notation is a specific way of writing numbers as a product of a number between 1 and 10 and a power of 10, while exponential notation is a more general way of writing numbers as a product of a base and an exponent.

Q: How do I convert a number from scientific notation to standard notation?

A: To convert a number from scientific notation to standard notation, you need to multiply the number by 10 raised to the power of the exponent. For example, $3 \times 10^4 = 30,000$.

Q: What are some common mistakes to avoid when working with scientific notation?

A: Some common mistakes to avoid when working with scientific notation include:

  • Forgetting to move the decimal point when converting a number to scientific notation
  • Forgetting to add or subtract the exponents when multiplying or dividing numbers in scientific notation
  • Not following the order of operations (PEMDAS)
  • Not using the correct notation for very large or very small numbers

Q: How do I use scientific notation in real-world applications?

A: Scientific notation is used in a wide range of real-world applications, including:

  • Calculating distances and velocities in physics and astronomy
  • Measuring the size of molecules and atoms in chemistry
  • Representing very large or very small numbers in finance and economics
  • Calculating the area and volume of complex shapes in engineering and architecture

Conclusion

Evaluating expressions in scientific notation can be a challenging task, but with practice and patience, you can become proficient in this area. By following the rules and guidelines outlined in this article, you can confidently evaluate expressions in scientific notation and apply this knowledge to real-world applications.

Frequently Asked Questions

  • Q: What is scientific notation? A: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10.
  • Q: How do I convert a number to scientific notation? A: To convert a number to scientific notation, you need to move the decimal point to the left or right until you have a number between 1 and 10.
  • Q: What is the rule for multiplying numbers in scientific notation? A: When multiplying numbers in scientific notation, you multiply the numbers and add the exponents.

Further Reading

  • Scientific Notation: A Guide to Understanding and Evaluating Expressions
  • Exponents: A Comprehensive Guide to Understanding and Applying the Rules
  • Multiplying Numbers in Scientific Notation: A Step-by-Step Guide

References

  • [1] "Scientific Notation" by Math Open Reference
  • [2] "Exponents" by Khan Academy
  • [3] "Multiplying Numbers in Scientific Notation" by IXL