Evaluate The Expression: ${ (-5)^7 \div (-5)^3 = }$

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Understanding Exponents and Their Properties

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In mathematics, exponents are a shorthand way of representing repeated multiplication. The expression $a^b$ means $a$ multiplied by itself $b$ times. For example, $2^3$ is equal to $2 \times 2 \times 2 = 8$. Exponents are a fundamental concept in mathematics, and understanding their properties is crucial for evaluating complex expressions.

The Power of a Power Property

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One of the most important properties of exponents is the power of a power property, which states that $(a^m)^n = a^{m \times n}$. This property allows us to simplify expressions by combining exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

The Product of Powers Property

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Another important property of exponents is the product of powers property, which states that $a^m \times a^n = a^{m + n}$. This property allows us to simplify expressions by combining exponents. For example, $2^3 \times 2^4 = 2^{3 + 4} = 2^7$.

The Quotient of Powers Property

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The quotient of powers property states that $\frac{a^m}{a^n} = a^{m - n}$. This property allows us to simplify expressions by combining exponents. For example, $\frac{2^5}{2^2} = 2^{5 - 2} = 2^3$.

Evaluating the Expression: $(-5)^7 \div (-5)^3$

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Now that we have a solid understanding of exponents and their properties, let's evaluate the expression $(-5)^7 \div (-5)^3$. To do this, we can use the quotient of powers property, which states that $\frac{a^m}{a^n} = a^{m - n}$.

Step 1: Simplify the Exponents

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Using the quotient of powers property, we can simplify the expression as follows:

$$(-5)^7 \div (-5)^3 = (-5)^{7 - 3} = (-5)^4$$

Step 2: Evaluate the Expression

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Now that we have simplified the expression, we can evaluate it by calculating the value of $(-5)^4$. To do this, we can use the fact that $(-a)^n = a^n$ when $n$ is even.

$$(-5)^4 = 5^4 = 625$$

Conclusion

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In conclusion, evaluating the expression $(-5)^7 \div (-5)^3$ requires a solid understanding of exponents and their properties. By using the quotient of powers property, we can simplify the expression and evaluate it to find the final answer.

Common Mistakes to Avoid

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When evaluating expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not simplifying the exponents: Failing to simplify the exponents can lead to incorrect answers.
• Not using the correct property: Using the wrong property can lead to incorrect answers.
• Not evaluating the expression correctly: Failing to evaluate the expression correctly can lead to incorrect answers.

Real-World Applications

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Exponents and their properties have many real-world applications. Here are a few examples:

• 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 systems.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.

Final Thoughts

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Evaluating expressions with exponents requires a solid understanding of exponents and their properties. By using the quotient of powers property and simplifying the exponents, we can evaluate complex expressions and find the final answer. Remember to avoid common mistakes and use real-world applications to reinforce your understanding of exponents and their properties.

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Frequently Asked Questions

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Q: What is the power of a power property?

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A: The power of a power property states that $(a^m)^n = a^{m \times n}$. This property allows us to simplify expressions by combining exponents.

Q: How do I simplify an expression with exponents?

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

1. Simplify the exponents by combining them using the power of a power property.
2. Use the product of powers property to combine exponents.
3. Use the quotient of powers property to simplify expressions.

Q: What is the product of powers property?

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A: The product of powers property states that $a^m \times a^n = a^{m + n}$. This property allows us to simplify expressions by combining exponents.

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

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A: To evaluate an expression with a negative exponent, you can use the following steps:

1. Simplify the expression by combining exponents.
2. Use the fact that $(-a)^n = a^n$ when $n$ is even.
3. Evaluate the expression to find the final answer.

Q: What is the quotient of powers property?

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A: The quotient of powers property states that $\frac{a^m}{a^n} = a^{m - n}$. This property allows us to simplify expressions by combining exponents.

Q: How do I simplify an expression with a fraction?

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A: To simplify an expression with a fraction, you can use the following steps:

1. Simplify the fraction by combining the numerator and denominator.
2. Use the quotient of powers property to simplify the expression.
3. Evaluate the expression to find the final answer.

Q: What are some common mistakes to avoid when evaluating expressions with exponents?

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A: Some common mistakes to avoid when evaluating expressions with exponents include:

• Not simplifying the exponents
• Not using the correct property
• Not evaluating the expression correctly

Q: How do I apply exponents in real-world situations?

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A: Exponents are used in many real-world situations, 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 systems.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.

Q: What are some tips for mastering exponents?

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A: Some tips for mastering exponents include:

• Practice, practice, practice: The more you practice evaluating expressions with exponents, the more comfortable you will become with the concepts.
• Use real-world examples: Using real-world examples can help you understand how exponents are used in different situations.
• Review and practice regularly: Reviewing and practicing exponents regularly can help you retain the information and build your skills.

Conclusion

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Evaluating expressions with exponents requires a solid understanding of exponents and their properties. By using the power of a power property, the product of powers property, and the quotient of powers property, you can simplify expressions and evaluate them to find the final answer. Remember to avoid common mistakes and use real-world applications to reinforce your understanding of exponents and their properties.