What Is The Product Of $\left(5^{-1}\right)\left(5^{-3}\right$\]?A. $\frac{1}{625}$ B. $\frac{1}{25}$ C. 25 D. 125

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Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. The exponentiation operation is denoted by a caret (^) or an exponentiation operator, and it is used to express a number raised to a power. For instance, 2^3 means 2 multiplied by itself three times, which equals 8. In this article, we will explore the concept of exponents and their applications, particularly in the context of the given problem: (5−1)(5−3)\left(5^{-1}\right)\left(5^{-3}\right).

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, 2^3 can be written as 2 × 2 × 2, which equals 8. The exponent, in this case, is 3, and it represents the number of times the base (2) is multiplied by itself. Exponents can be positive or negative, and they can also be fractional or decimal.

Properties of Exponents

Exponents have several properties that make them useful in mathematics. Some of the key properties of exponents include:

  • Product of Powers: When multiplying two numbers with the same base, we add their exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
  • Power of a Power: When raising a power to another power, we multiply the exponents. For example, (23)4 = 2^(3×4) = 2^12.
  • Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, 2^0 = 1.
  • Negative Exponent: A negative exponent represents the reciprocal of the base raised to the positive exponent. For example, 2^(-3) = 1/2^3.

Solving the Given Problem

Now that we have a good understanding of exponents and their properties, let's apply this knowledge to solve the given problem: (5−1)(5−3)\left(5^{-1}\right)\left(5^{-3}\right).

Using the property of negative exponents, we can rewrite the expression as:

(5−1)(5−3)=151×153\left(5^{-1}\right)\left(5^{-3}\right) = \frac{1}{5^1} \times \frac{1}{5^3}

Now, we can apply the product of powers property to simplify the expression:

151×153=151+3=154\frac{1}{5^1} \times \frac{1}{5^3} = \frac{1}{5^{1+3}} = \frac{1}{5^4}

Finally, we can evaluate the expression by calculating the value of 545^4:

54=5×5×5×5=6255^4 = 5 \times 5 \times 5 \times 5 = 625

Therefore, the value of (5−1)(5−3)\left(5^{-1}\right)\left(5^{-3}\right) is 1625\frac{1}{625}.

Conclusion

In conclusion, exponents are a fundamental concept in mathematics that represents repeated multiplication of a number. Understanding the properties of exponents, such as the product of powers, power of a power, zero exponent, and negative exponent, is crucial in solving problems involving exponents. By applying these properties, we can simplify complex expressions and arrive at the correct solution. In this article, we solved the given problem (5−1)(5−3)\left(5^{-1}\right)\left(5^{-3}\right) using the properties of exponents and arrived at the correct solution: 1625\frac{1}{625}.

Common Mistakes to Avoid

When working with exponents, it's essential to avoid common mistakes that can lead to incorrect solutions. Some of the common mistakes to avoid include:

  • Incorrect application of exponent properties: Make sure to apply the correct exponent properties, such as the product of powers, power of a power, zero exponent, and negative exponent.
  • Incorrect evaluation of expressions: Double-check your calculations to ensure that you are evaluating the expression correctly.
  • Lack of understanding of exponent properties: Make sure to understand the properties of exponents before applying them to solve problems.

Real-World Applications of Exponents

Exponents have numerous real-world applications in various fields, including:

  • Science: Exponents are used to represent the growth or decay of populations, chemical reactions, and physical phenomena.
  • Engineering: Exponents are used to represent the scaling of physical systems, such as the size of buildings, bridges, and electronic circuits.
  • Finance: Exponents are used to represent the growth or decay of investments, such as stocks, bonds, and mutual funds.

Final Thoughts

Introduction

In our previous article, we explored the concept of exponents and their applications in mathematics. We also solved a problem involving exponents and arrived at the correct solution. In this article, we will address some frequently asked questions about exponents and provide answers to help you better understand this concept.

Q1: What is the difference between a base and an exponent?

A1: The base is the number being multiplied by itself, and the exponent is the number of times the base is multiplied by itself. For example, in the expression 2^3, the base is 2 and the exponent is 3.

Q2: How do I simplify an expression with exponents?

A2: To simplify an expression with exponents, you can use the following steps:

  1. Identify the base and exponent in the expression.
  2. Apply the exponent properties, such as the product of powers, power of a power, zero exponent, and negative exponent.
  3. Simplify the expression by evaluating the exponent.

Q3: What is the rule for multiplying exponents with the same base?

A3: When multiplying exponents with the same base, you add the exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.

Q4: What is the rule for raising a power to another power?

A4: When raising a power to another power, you multiply the exponents. For example, (23)4 = 2^(3×4) = 2^12.

Q5: What is the value of 0 raised to any power?

A5: Any non-zero number raised to the power of zero is equal to 1. For example, 2^0 = 1.

Q6: What is the value of a negative exponent?

A6: A negative exponent represents the reciprocal of the base raised to the positive exponent. For example, 2^(-3) = 1/2^3.

Q7: How do I evaluate an expression with exponents?

A7: To evaluate an expression with exponents, you can use the following steps:

  1. Identify the base and exponent in the expression.
  2. Apply the exponent properties, such as the product of powers, power of a power, zero exponent, and negative exponent.
  3. Simplify the expression by evaluating the exponent.

Q8: What are some common mistakes to avoid when working with exponents?

A8: Some common mistakes to avoid when working with exponents include:

  • Incorrect application of exponent properties
  • Incorrect evaluation of expressions
  • Lack of understanding of exponent properties

Q9: What are some real-world applications of exponents?

A9: Exponents have numerous real-world applications in various fields, including:

  • Science: Exponents are used to represent the growth or decay of populations, chemical reactions, and physical phenomena.
  • Engineering: Exponents are used to represent the scaling of physical systems, such as the size of buildings, bridges, and electronic circuits.
  • Finance: Exponents are used to represent the growth or decay of investments, such as stocks, bonds, and mutual funds.

Q10: How can I practice working with exponents?

A10: You can practice working with exponents by:

  • Solving problems involving exponents
  • Using online resources, such as calculators and worksheets
  • Working with a tutor or teacher to understand the concept of exponents

Conclusion

In conclusion, exponents are a fundamental concept in mathematics that represents repeated multiplication of a number. Understanding the properties of exponents, such as the product of powers, power of a power, zero exponent, and negative exponent, is crucial in solving problems involving exponents. By applying these properties, we can simplify complex expressions and arrive at the correct solution. In this article, we addressed some frequently asked questions about exponents and provided answers to help you better understand this concept.