Simplify. Express Your Answer Using A Single Exponent. ( Q − 8 ) 5 \left(q^{-8}\right)^5 ( Q − 8 ) 5 □ \square □

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Understanding Exponents

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will focus on simplifying expressions with exponents, specifically the expression (q−8)5\left(q^{-8}\right)^5.

The Power of a Power Rule

The power of a power rule is a fundamental rule in exponentiation that states: (am)n=am⋅n(a^m)^n = a^{m \cdot n}. This rule allows us to simplify expressions with exponents by multiplying the exponents.

Applying the Power of a Power Rule

To simplify the expression (q−8)5\left(q^{-8}\right)^5, we can apply the power of a power rule. We multiply the exponents, −8-8 and 55, to get:

(q−8)5=q−8⋅5=q−40\left(q^{-8}\right)^5 = q^{-8 \cdot 5} = q^{-40}

Simplifying Negative Exponents

Negative exponents can be simplified by taking the reciprocal of the base and changing the sign of the exponent. In this case, we can rewrite q−40q^{-40} as:

1q40\frac{1}{q^{40}}

Understanding the Concept of Reciprocal

The reciprocal of a number is simply 1 divided by that number. In the case of q−40q^{-40}, the reciprocal is 1q40\frac{1}{q^{40}}. This concept is essential in simplifying expressions with negative exponents.

Real-World Applications of Exponents

Exponents have numerous real-world applications, including:

  • Finance: Exponents are used to calculate compound interest and investment returns.
  • Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical phenomena.
  • Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, simplifying expressions with exponents is a fundamental concept in mathematics. By applying the power of a power rule and understanding the concept of reciprocal, we can simplify expressions with negative exponents. The power of exponents has numerous real-world applications, making it an essential tool in various fields.

Additional Examples

Here are some additional examples of simplifying expressions with exponents:

  • (q12)3=q12⋅3=q36\left(q^{12}\right)^3 = q^{12 \cdot 3} = q^{36}
  • (q−3)4=q−3⋅4=q−12=1q12\left(q^{-3}\right)^4 = q^{-3 \cdot 4} = q^{-12} = \frac{1}{q^{12}}
  • (q5)−2=q5⋅−2=q−10=1q10\left(q^{5}\right)^{-2} = q^{5 \cdot -2} = q^{-10} = \frac{1}{q^{10}}

Practice Problems

Here are some practice problems to help you reinforce your understanding of simplifying expressions with exponents:

  • Simplify (q15)2\left(q^{15}\right)^2
  • Simplify (q−9)3\left(q^{-9}\right)^3
  • Simplify (q7)−4\left(q^{7}\right)^{-4}

Answer Key

Here are the answers to the practice problems:

  • (q15)2=q15⋅2=q30\left(q^{15}\right)^2 = q^{15 \cdot 2} = q^{30}
  • (q−9)3=q−9⋅3=q−27=1q27\left(q^{-9}\right)^3 = q^{-9 \cdot 3} = q^{-27} = \frac{1}{q^{27}}
  • (q7)−4=q7⋅−4=q−28=1q28\left(q^{7}\right)^{-4} = q^{7 \cdot -4} = q^{-28} = \frac{1}{q^{28}}
    Simplify Expressions with Exponents: Q&A =====================================

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about simplifying expressions with exponents.

Q: What is the power of a power rule?

A: The power of a power rule is a fundamental rule in exponentiation that states: (am)n=am⋅n(a^m)^n = a^{m \cdot n}. This rule allows us to simplify expressions with exponents by multiplying the exponents.

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

A: To simplify an expression with a negative exponent, you can take the reciprocal of the base and change the sign of the exponent. For example, q−40q^{-40} can be rewritten as 1q40\frac{1}{q^{40}}.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power. For example, q40q^{40} indicates that qq is raised to the power of 40, while q−40q^{-40} indicates that the reciprocal of qq is raised to the power of 40.

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents by applying the power of a power rule. For example, (q12)3\left(q^{12}\right)^3 can be simplified as q12⋅3=q36q^{12 \cdot 3} = q^{36}.

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

A: To simplify an expression with a fractional exponent, you can rewrite it as a product of two exponents. For example, q12q^{\frac{1}{2}} can be rewritten as q\sqrt{q}.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains an exponent, while a polynomial expression is an expression that contains only variables and coefficients. For example, q40q^{40} is an exponential expression, while q+2q + 2 is a polynomial expression.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent by applying the power of a power rule. For example, (qx)y\left(q^x\right)^y can be simplified as qx⋅yq^{x \cdot y}.

Q: How do I simplify an expression with a negative base and a positive exponent?

A: To simplify an expression with a negative base and a positive exponent, you can rewrite the base as a positive number and change the sign of the exponent. For example, (−q)40(-q)^{40} can be rewritten as q40q^{40}.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by setting the base equal to 1. For example, q0=1q^0 = 1.

Q: How do I simplify an expression with a variable base and a zero exponent?

A: To simplify an expression with a variable base and a zero exponent, you can set the base equal to 1. For example, q0=1q^0 = 1.

Conclusion

In conclusion, simplifying expressions with exponents is a fundamental concept in mathematics. By understanding the power of a power rule, negative exponents, and fractional exponents, you can simplify a wide range of expressions. Remember to always apply the rules of exponentiation and to be careful when simplifying expressions with variables in the exponents.

Additional Resources

For more information on simplifying expressions with exponents, check out the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Practice Problems

Here are some practice problems to help you reinforce your understanding of simplifying expressions with exponents:

  • Simplify (q15)2\left(q^{15}\right)^2
  • Simplify (q−9)3\left(q^{-9}\right)^3
  • Simplify (q7)−4\left(q^{7}\right)^{-4}
  • Simplify q0q^0
  • Simplify (−q)40(-q)^{40}

Answer Key

Here are the answers to the practice problems:

  • (q15)2=q15⋅2=q30\left(q^{15}\right)^2 = q^{15 \cdot 2} = q^{30}
  • (q−9)3=q−9⋅3=q−27=1q27\left(q^{-9}\right)^3 = q^{-9 \cdot 3} = q^{-27} = \frac{1}{q^{27}}
  • (q7)−4=q7⋅−4=q−28=1q28\left(q^{7}\right)^{-4} = q^{7 \cdot -4} = q^{-28} = \frac{1}{q^{28}}
  • q0=1q^0 = 1
  • (−q)40=q40(-q)^{40} = q^{40}