Match Each Expression On The Left To Its Product.$[ \begin{array}{|c|c|c|c|c|} \hline & 5.02 & 502 & 50.2 & 0.502 \ \hline 50.2 \times \frac{1}{10^2} & \square & \square & \square & \square \ \hline 5.02 \times 10^0 & \square & \square &

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Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us evaluate and manipulate complex expressions with ease. When dealing with exponential expressions, it's essential to understand the rules of exponents and how to apply them to simplify expressions. In this article, we will explore how to match each expression on the left to its product, focusing on the given expressions involving powers of 10.

Understanding Exponents

Before we dive into the problem, let's quickly review the basics of exponents. An exponent is a small number that is placed above and to the right of a number, indicating how many times the base number should be multiplied by itself. For example, in the expression 232^3, the exponent 3 indicates that the base number 2 should be multiplied by itself 3 times: 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8.

Simplifying Exponential Expressions

Now that we have a basic understanding of exponents, let's move on to simplifying exponential expressions. When simplifying exponential expressions, we need to apply the rules of exponents, which include:

  • Product of Powers Rule: When multiplying two exponential expressions with the same base, we add the exponents. For example, 23×24=23+4=272^3 \times 2^4 = 2^{3+4} = 2^7
  • Power of a Power Rule: When raising an exponential expression to a power, we multiply the exponents. For example, (23)4=23×4=212(2^3)^4 = 2^{3 \times 4} = 2^{12}
  • Quotient of Powers Rule: When dividing two exponential expressions with the same base, we subtract the exponents. For example, 2523=25−3=22\frac{2^5}{2^3} = 2^{5-3} = 2^2

Applying the Rules of Exponents

Now that we have reviewed the rules of exponents, let's apply them to the given expressions.

50.2 × 1/10^2

To simplify this expression, we need to apply the quotient of powers rule. Since the base is 10, we can rewrite the expression as:

50.2 × 10^(-2)

Using the quotient of powers rule, we can simplify this expression by subtracting the exponents:

50.2 × 10^(-2) = 50.2 ÷ 10^2 = 50.2 ÷ 100 = 0.502

Therefore, the product of 50.2 × 1/10^2 is 0.502.

5.02 × 10^0

To simplify this expression, we need to apply the product of powers rule. Since the exponent is 0, we can rewrite the expression as:

5.02 × 10^0 = 5.02

Using the product of powers rule, we can simplify this expression by adding the exponents:

5.02 × 10^0 = 5.02

Therefore, the product of 5.02 × 10^0 is 5.02.

502 × 1/10^2

To simplify this expression, we need to apply the quotient of powers rule. Since the base is 10, we can rewrite the expression as:

502 × 10^(-2)

Using the quotient of powers rule, we can simplify this expression by subtracting the exponents:

502 × 10^(-2) = 502 ÷ 10^2 = 502 ÷ 100 = 5.02

Therefore, the product of 502 × 1/10^2 is 5.02.

0.502 × 10^2

To simplify this expression, we need to apply the product of powers rule. Since the exponent is 2, we can rewrite the expression as:

0.502 × 10^2 = 0.502 × 100 = 50.2

Using the product of powers rule, we can simplify this expression by adding the exponents:

0.502 × 10^2 = 50.2

Therefore, the product of 0.502 × 10^2 is 50.2.

Conclusion

In this article, we have explored how to match each expression on the left to its product, focusing on the given expressions involving powers of 10. By applying the rules of exponents, we were able to simplify each expression and find its product. We hope that this guide has been helpful in understanding how to simplify exponential expressions and apply the rules of exponents.

Discussion

  • What are some common mistakes to avoid when simplifying exponential expressions?
  • How can we apply the rules of exponents to simplify complex expressions?
  • What are some real-world applications of simplifying exponential expressions?

References

Additional Resources

  • [1] Exponents and Exponential Functions (Khan Academy)
  • [2] Exponents and Exponential Functions (Mathway)
  • [3] Simplifying Exponential Expressions (Purplemath)
    Frequently Asked Questions: Simplifying Exponential Expressions ====================================================================

Q: What is the difference between a base and an exponent?

A: In an exponential expression, the base is the number being raised to a power, and the exponent is the number that indicates how many times the base should be multiplied by itself. For example, in the expression 232^3, the base is 2 and the exponent is 3.

Q: How do I simplify an exponential expression with a negative exponent?

A: To simplify an exponential expression with a negative exponent, you can rewrite the expression as a fraction. For example, 2−32^{-3} can be rewritten as 123\frac{1}{2^3}. This is because 2−32^{-3} means 1 divided by 232^3.

Q: What is the rule for multiplying exponential expressions with the same base?

A: When multiplying exponential expressions with the same base, you add the exponents. For example, 23×24=23+4=272^3 \times 2^4 = 2^{3+4} = 2^7.

Q: What is the rule for dividing exponential expressions with the same base?

A: When dividing exponential expressions with the same base, you subtract the exponents. For example, 2523=25−3=22\frac{2^5}{2^3} = 2^{5-3} = 2^2.

Q: How do I simplify an exponential expression with a zero exponent?

A: An exponential expression with a zero exponent is equal to 1. For example, 20=12^0 = 1.

Q: Can I simplify an exponential expression with a fractional exponent?

A: Yes, you can simplify an exponential expression with a fractional exponent. For example, 2122^{\frac{1}{2}} can be rewritten as 2\sqrt{2}.

Q: How do I simplify an exponential expression with a negative fractional exponent?

A: To simplify an exponential expression with a negative fractional exponent, you can rewrite the expression as a fraction. For example, 2−122^{-\frac{1}{2}} can be rewritten as 12\frac{1}{\sqrt{2}}.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

  • Forgetting to apply the rules of exponents
  • Not simplifying the expression correctly
  • Not checking the answer for accuracy

Q: How can I apply the rules of exponents to simplify complex expressions?

A: To apply the rules of exponents to simplify complex expressions, you can follow these steps:

  1. Identify the base and exponent in the expression
  2. Apply the rules of exponents to simplify the expression
  3. Check the answer for accuracy

Q: What are some real-world applications of simplifying exponential expressions?

A: Simplifying exponential expressions has many real-world applications, including:

  • Calculating interest rates and investments
  • Modeling population growth and decay
  • Analyzing data and statistics

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponential expressions. We hope that this guide has been helpful in understanding how to simplify exponential expressions and apply the rules of exponents.

Discussion

  • What are some other common mistakes to avoid when simplifying exponential expressions?
  • How can we apply the rules of exponents to simplify complex expressions in real-world applications?
  • What are some other real-world applications of simplifying exponential expressions?

References

Additional Resources

  • [1] Exponents and Exponential Functions (Khan Academy)
  • [2] Exponents and Exponential Functions (Mathway)
  • [3] Simplifying Exponential Expressions (Purplemath)