Which Expression Is Equivalent To ( 2 3 ) − 5 \left(2^3\right)^{-5} ( 2 3 ) − 5 ?

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Understanding Exponents and Their Properties

Exponents are a fundamental concept in mathematics, and they play a crucial role in algebra, geometry, and other branches of mathematics. In this article, we will explore the concept of exponents and their properties, with a focus on understanding which expression is equivalent to $\left(2^3\right)^{-5}$.

What are Exponents?

Exponents are a shorthand way of expressing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed." It means that 2 is multiplied by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8$.

Properties of Exponents

Exponents have several properties that are essential to understanding how they work. Some of the key properties of exponents include:

• Product of Powers Property: When multiplying two powers with the same base, we add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
• Power of a Power Property: When raising a power to another power, we multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.
• Zero Exponent Property: Any non-zero number raised to the power of 0 is equal to 1. For example, $2^0 = 1$.
• Negative Exponent Property: A negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Applying Exponent Properties to $\left(2^3\right)^{-5}$

Now that we have a solid understanding of exponents and their properties, let's apply these concepts to the expression $\left(2^3\right)^{-5}$.

Using the Power of a Power Property

Using the power of a power property, we can rewrite $\left(2^3\right)^{-5}$ as $2^{3 \times -5}$. This simplifies to $2^{-15}$.

Using the Negative Exponent Property

Using the negative exponent property, we can rewrite $2^{-15}$ as $\frac{1}{2^{15}}$.

Simplifying the Expression

Now that we have rewritten the expression as $\frac{1}{2^{15}}$, we can simplify it further. Since $2^{15}$ is a large number, we can rewrite it as a product of smaller numbers. For example, $2^{15} = 2^5 \times 2^5 \times 2^5 = 32 \times 32 \times 32 = 32768$.

Final Answer

Therefore, the expression $\left(2^3\right)^{-5}$ is equivalent to $\frac{1}{32768}$.

Conclusion

In this article, we explored the concept of exponents and their properties, with a focus on understanding which expression is equivalent to $\left(2^3\right)^{-5}$. We applied the power of a power property and the negative exponent property to simplify the expression and arrive at the final answer. We hope that this article has provided a clear and concise explanation of the concept of exponents and their properties, and has helped readers to better understand this important mathematical concept.

Frequently Asked Questions

• What is the definition of an exponent? An exponent is a shorthand way of expressing repeated multiplication.
• What are the properties of exponents? The properties of exponents include the product of powers property, the power of a power property, the zero exponent property, and the negative exponent property.
• How do you simplify an expression with a negative exponent? To simplify an expression with a negative exponent, you can use the negative exponent property to rewrite it as a fraction.
• What is the final answer to the expression $\left(2^3\right)^{-5}$? The final answer to the expression $\left(2^3\right)^{-5}$ is $\frac{1}{32768}$.

Further Reading

• Exponents and Their Properties: This article provides a comprehensive overview of exponents and their properties.
• Simplifying Expressions with Exponents: This article provides tips and tricks for simplifying expressions with exponents.
• Negative Exponents: This article provides a detailed explanation of negative exponents and how to simplify expressions with them.

References

• "Algebra" by Michael Artin: This book provides a comprehensive introduction to algebra, including exponents and their properties.
• "Mathematics for the Nonmathematician" by Morris Kline: This book provides a clear and concise introduction to mathematics, including exponents and their properties.
• "The Art of Mathematics" by Tom M. Apostol: This book provides a comprehensive introduction to mathematics, including exponents and their properties.
  

Understanding Exponents and Their Properties

Exponents are a fundamental concept in mathematics, and they play a crucial role in algebra, geometry, and other branches of mathematics. In this article, we will explore the concept of exponents and their properties, with a focus on answering some of the most frequently asked questions about exponents.

Q&A: Exponents and Their Properties

Q: What is the definition of an exponent?

A: An exponent is a shorthand way of expressing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed." It means that 2 is multiplied by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8$.

Q: What are the properties of exponents?

A: The properties of exponents include the product of powers property, the power of a power property, the zero exponent property, and the negative exponent property.

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

A: To simplify an expression with a negative exponent, you can use the negative exponent property to rewrite it as a fraction. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the final answer to the expression $\left(2^3\right)^{-5}$?

A: The final answer to the expression $\left(2^3\right)^{-5}$ is $\frac{1}{32768}$.

Q: How do you apply the power of a power property?

A: To apply the power of a power property, you multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

Q: What is the zero exponent property?

A: The zero exponent property states that any non-zero number raised to the power of 0 is equal to 1. For example, $2^0 = 1$.

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

A: To simplify an expression with a zero exponent, you can use the zero exponent property to rewrite it as 1. For example, $2^0 = 1$.

Q: What is the negative exponent property?

A: The negative exponent property states that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

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

A: To simplify an expression with a negative exponent, you can use the negative exponent property to rewrite it as a fraction. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Conclusion

In this article, we have explored the concept of exponents and their properties, with a focus on answering some of the most frequently asked questions about exponents. We hope that this article has provided a clear and concise explanation of the concept of exponents and their properties, and has helped readers to better understand this important mathematical concept.

Frequently Asked Questions

• What is the definition of an exponent? An exponent is a shorthand way of expressing repeated multiplication.
• What are the properties of exponents? The properties of exponents include the product of powers property, the power of a power property, the zero exponent property, and the negative exponent property.
• How do you simplify an expression with a negative exponent? To simplify an expression with a negative exponent, you can use the negative exponent property to rewrite it as a fraction.
• What is the final answer to the expression $\left(2^3\right)^{-5}$? The final answer to the expression $\left(2^3\right)^{-5}$ is $\frac{1}{32768}$.

Further Reading

• Exponents and Their Properties: This article provides a comprehensive overview of exponents and their properties.
• Simplifying Expressions with Exponents: This article provides tips and tricks for simplifying expressions with exponents.
• Negative Exponents: This article provides a detailed explanation of negative exponents and how to simplify expressions with them.

References

• "Algebra" by Michael Artin: This book provides a comprehensive introduction to algebra, including exponents and their properties.
• "Mathematics for the Nonmathematician" by Morris Kline: This book provides a clear and concise introduction to mathematics, including exponents and their properties.
• "The Art of Mathematics" by Tom M. Apostol: This book provides a comprehensive introduction to mathematics, including exponents and their properties.