Evaluate The Expression:$\[ \frac{6 \times 10^{-2}}{8 \times 10^7} \\]

by ADMIN 71 views

Introduction

In mathematics, we often encounter expressions that involve exponents and fractions. Evaluating such expressions requires a clear understanding of the rules of exponents and fractions. In this article, we will evaluate the expression 6×10−28×107\frac{6 \times 10^{-2}}{8 \times 10^7} and explore the concepts that are involved in its evaluation.

Understanding Exponents and Fractions

Before we dive into the evaluation of the expression, let's briefly review the rules of exponents and fractions. Exponents are a shorthand way of writing repeated multiplication. For example, 232^3 means 2×2×22 \times 2 \times 2. When we have a fraction with exponents, we can simplify it by applying the rules of exponents.

Evaluating the Expression

To evaluate the expression 6×10−28×107\frac{6 \times 10^{-2}}{8 \times 10^7}, we need to follow the order of operations (PEMDAS):

  1. Multiply the numbers in the numerator and denominator.
  2. Simplify the fraction by applying the rules of exponents.

Let's start by multiplying the numbers in the numerator and denominator:

6×10−28×107=68×10−2107\frac{6 \times 10^{-2}}{8 \times 10^7} = \frac{6}{8} \times \frac{10^{-2}}{10^7}

Simplifying the Fraction

Now, let's simplify the fraction by applying the rules of exponents. When we divide two numbers with the same base, we subtract the exponents:

10−2107=10−2−7=10−9\frac{10^{-2}}{10^7} = 10^{-2-7} = 10^{-9}

Applying the Rules of Exponents

Now, let's apply the rules of exponents to the fraction:

68×10−9=34×10−9\frac{6}{8} \times 10^{-9} = \frac{3}{4} \times 10^{-9}

Evaluating the Final Expression

Finally, let's evaluate the final expression:

34×10−9=0.75×10−9\frac{3}{4} \times 10^{-9} = 0.75 \times 10^{-9}

Conclusion

In this article, we evaluated the expression 6×10−28×107\frac{6 \times 10^{-2}}{8 \times 10^7} and explored the concepts that are involved in its evaluation. We learned how to simplify fractions with exponents and how to apply the rules of exponents to evaluate expressions.

Importance of Evaluating Expressions

Evaluating expressions is an essential skill in mathematics and science. It helps us to understand the behavior of physical systems and to make predictions about the future. In this article, we saw how evaluating the expression 6×10−28×107\frac{6 \times 10^{-2}}{8 \times 10^7} can help us to understand the behavior of a physical system.

Real-World Applications

Evaluating expressions has many real-world applications. For example, in physics, we use expressions to describe the motion of objects. In engineering, we use expressions to design and optimize systems. In finance, we use expressions to model the behavior of financial markets.

Tips for Evaluating Expressions

Here are some tips for evaluating expressions:

  • Follow the order of operations: PEMDAS is a mnemonic device that helps us to remember the order of operations.
  • Simplify fractions: Simplifying fractions can help us to evaluate expressions more easily.
  • Apply the rules of exponents: The rules of exponents are essential for evaluating expressions with exponents.
  • Use a calculator: If you are unsure about the value of an expression, you can use a calculator to evaluate it.

Final Thoughts

Evaluating expressions is an essential skill in mathematics and science. It helps us to understand the behavior of physical systems and to make predictions about the future. In this article, we evaluated the expression 6×10−28×107\frac{6 \times 10^{-2}}{8 \times 10^7} and explored the concepts that are involved in its evaluation. We learned how to simplify fractions with exponents and how to apply the rules of exponents to evaluate expressions.

Introduction

In our previous article, we evaluated the expression 6×10−28×107\frac{6 \times 10^{-2}}{8 \times 10^7} and explored the concepts that are involved in its evaluation. In this article, we will answer some frequently asked questions about evaluating expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify fractions with exponents?

A: To simplify fractions with exponents, we need to apply the rules of exponents. When we divide two numbers with the same base, we subtract the exponents. For example:

10−2107=10−2−7=10−9\frac{10^{-2}}{10^7} = 10^{-2-7} = 10^{-9}

Q: What is the rule for multiplying fractions with exponents?

A: When we multiply fractions with exponents, we add the exponents. For example:

2×1033×102=23×103+2=23×105\frac{2 \times 10^3}{3 \times 10^2} = \frac{2}{3} \times 10^{3+2} = \frac{2}{3} \times 10^5

Q: How do I evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, we need to apply the rule that a−n=1ana^{-n} = \frac{1}{a^n}. For example:

110−3=103\frac{1}{10^{-3}} = 10^{3}

Q: What is the rule for dividing fractions with exponents?

A: When we divide fractions with exponents, we subtract the exponents. For example:

10−2107=10−2−7=10−9\frac{10^{-2}}{10^7} = 10^{-2-7} = 10^{-9}

Q: How do I evaluate expressions with scientific notation?

A: To evaluate expressions with scientific notation, we need to apply the rules of exponents. For example:

3.4×102÷2.1×103=3.42.1×102−3=3.42.1×10−13.4 \times 10^2 \div 2.1 \times 10^3 = \frac{3.4}{2.1} \times 10^{2-3} = \frac{3.4}{2.1} \times 10^{-1}

Q: What is the rule for adding and subtracting fractions with exponents?

A: When we add and subtract fractions with exponents, we need to have the same base and exponent. For example:

2×1033×102+4×1035×102=23×105+45×105=23×105+45×105\frac{2 \times 10^3}{3 \times 10^2} + \frac{4 \times 10^3}{5 \times 10^2} = \frac{2}{3} \times 10^5 + \frac{4}{5} \times 10^5 = \frac{2}{3} \times 10^5 + \frac{4}{5} \times 10^5

Conclusion

Evaluating expressions is an essential skill in mathematics and science. In this article, we answered some frequently asked questions about evaluating expressions. We learned how to simplify fractions with exponents, how to apply the rules of exponents, and how to evaluate expressions with scientific notation.

Tips for Evaluating Expressions

Here are some tips for evaluating expressions:

  • Follow the order of operations: PEMDAS is a mnemonic device that helps us to remember the order of operations.
  • Simplify fractions: Simplifying fractions can help us to evaluate expressions more easily.
  • Apply the rules of exponents: The rules of exponents are essential for evaluating expressions with exponents.
  • Use a calculator: If you are unsure about the value of an expression, you can use a calculator to evaluate it.

Final Thoughts

Evaluating expressions is an essential skill in mathematics and science. It helps us to understand the behavior of physical systems and to make predictions about the future. In this article, we answered some frequently asked questions about evaluating expressions and provided some tips for evaluating expressions.