Evaluate The Expression $m^2 \cdot M^5 \cdot M^{-3}$. Enter The Correct Answer In The Box, Ensuring That Your Answer Is In Simplest Form.

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will focus on simplifying exponential expressions, specifically the expression $m^2 \cdot m^5 \cdot m^{-3}$. To simplify this expression, we need to understand the rules of exponents and how to apply them.

The Rules of Exponents

There are several rules of exponents that we need to know to simplify the given expression. These rules are:

  • Product of Powers Rule: When multiplying two or more powers with the same base, we add the exponents. In other words, $a^m \cdot a^n = a^{m+n}$.
  • Power of a Power Rule: When raising a power to another power, we multiply the exponents. In other words, $(am)n = a^{m \cdot n}$.
  • Quotient of Powers Rule: When dividing two powers with the same base, we subtract the exponents. In other words, $\frac{am}{an} = a^{m-n}$.
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. In other words, $a^0 = 1$.

Simplifying the Expression

Now that we have a good understanding of the rules of exponents, let's simplify the expression $m^2 \cdot m^5 \cdot m^{-3}$. To do this, we will apply the product of powers rule, which states that when multiplying two or more powers with the same base, we add the exponents.

m2⋅m5⋅m−3=m2+5−3m^2 \cdot m^5 \cdot m^{-3} = m^{2+5-3}

Applying the Product of Powers Rule

Now, let's apply the product of powers rule to simplify the expression.

m2+5−3=m4m^{2+5-3} = m^{4}

The Final Answer

Therefore, the simplified form of the expression $m^2 \cdot m^5 \cdot m^{-3}$ is $m^4$.

Conclusion

In this article, we have learned how to simplify exponential expressions using the rules of exponents. We have applied the product of powers rule to simplify the expression $m^2 \cdot m^5 \cdot m^{-3}$ and arrived at the final answer of $m^4$. This is a fundamental concept in mathematics, and understanding it is crucial for solving problems in algebra and other branches of mathematics.

Frequently Asked Questions

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two or more powers with the same base, we add the exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the powers with the same base.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, apply the rules of exponents, such as the product of powers rule, the power of a power rule, and the quotient of powers rule.

Additional Resources

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

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

By following the steps outlined in this article and applying the rules of exponents, you will be able to simplify exponential expressions with ease. Remember to always check your work and verify your answers to ensure accuracy.

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Understanding Exponents and Exponential Functions

Exponents and exponential functions are fundamental concepts in mathematics, used to represent repeated multiplication of a number. In this article, we will answer some of the most frequently asked questions about exponents and exponential functions.

Q&A: Exponents and Exponential Functions

Q: What is an exponent?

A: An exponent is a small number that is written above and to the right of a number, indicating how many times the number should be multiplied by itself. For example, in the expression $2^3$, the exponent 3 indicates that 2 should be multiplied by itself 3 times.

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

A: An exponent is the number that is written above and to the right of a number, while a power is the result of raising a number to a certain exponent. For example, in the expression $2^3$, 3 is the exponent and 8 is the power.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two or more powers with the same base, we add the exponents. In other words, $a^m \cdot a^n = a^{m+n}$.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the powers with the same base. For example, in the expression $2^3 \cdot 2^4$, we add the exponents 3 and 4 to get $2^{3+4} = 2^7$.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, we multiply the exponents. In other words, $(am)n = a^{m \cdot n}$.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, simply multiply the exponents of the powers. For example, in the expression $(23)4$, we multiply the exponents 3 and 4 to get $2^{3 \cdot 4} = 2^{12}$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, we subtract the exponents. In other words, $\frac{am}{an} = a^{m-n}$.

Q: How do I apply the quotient of powers rule?

A: To apply the quotient of powers rule, simply subtract the exponents of the powers. For example, in the expression $\frac{25}{23}$, we subtract the exponents 5 and 3 to get $2^{5-3} = 2^2$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. In other words, $a^0 = 1$.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, apply the rules of exponents, such as the product of powers rule, the power of a power rule, and the quotient of powers rule.

Additional Resources

For more information on exponents and exponential functions, check out the following resources:

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

By following the steps outlined in this article and applying the rules of exponents, you will be able to simplify exponential expressions with ease. Remember to always check your work and verify your answers to ensure accuracy.

Common Mistakes to Avoid

When working with exponents and exponential functions, there are several common mistakes to avoid:

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying exponential expressions.
  • Not applying the rules of exponents correctly: Make sure to apply the rules of exponents correctly, such as the product of powers rule, the power of a power rule, and the quotient of powers rule.
  • Not checking your work: Make sure to check your work and verify your answers to ensure accuracy.

By avoiding these common mistakes, you will be able to simplify exponential expressions with ease and accuracy.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponents and exponential functions. We have covered the product of powers rule, the power of a power rule, the quotient of powers rule, and the zero exponent rule. We have also provided additional resources and common mistakes to avoid. By following the steps outlined in this article and applying the rules of exponents, you will be able to simplify exponential expressions with ease and accuracy.