Write An Expression That Includes A Product And A Power To A Power And Simplifies To M 7 N 6 M^7 N^6 M 7 N 6 .

Introduction

In algebra, we often encounter expressions that involve products and powers. These expressions can be simplified using various mathematical techniques. In this article, we will focus on writing an expression that includes a product and a power to a power and simplifies to $m^7 n^6$. We will explore the different ways to simplify such expressions and provide examples to illustrate the concepts.

Understanding Products and Powers

Before we dive into simplifying expressions, let's review the basics of products and powers.

• A product is the result of multiplying two or more numbers. For example, $2 \times 3 = 6$.
• A power is the result of raising a number to a certain exponent. For example, $2^3 = 8$.

Simplifying Expressions with Products and Powers

Now that we have reviewed the basics of products and powers, let's move on to simplifying expressions that involve both.

Example 1: Simplifying a Product of Powers

Suppose we want to simplify the expression $(m^2 n^3)^4$. To do this, we can use the power to a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

(m^2 n^3)^4 = m^{2 \cdot 4} n^{3 \cdot 4} = m^8 n^{12}

In this example, we raised the product of $m^2$ and $n^3$ to the power of 4, resulting in $m^8 n^{12}$.

Example 2: Simplifying a Power to a Power

Suppose we want to simplify the expression $(m^3 n^2)^2$. To do this, we can use the power to a power rule again.

(m^3 n^2)^2 = m^{3 \cdot 2} n^{2 \cdot 2} = m^6 n^4

In this example, we raised the product of $m^3$ and $n^2$ to the power of 2, resulting in $m^6 n^4$.

Example 3: Simplifying a Product of Powers with Different Exponents

Suppose we want to simplify the expression $(m^2 n^3)^4 (m^5 n^2)$. To do this, we can use the power to a power rule and the product rule, which states that $a^m \cdot a^n = a^{m + n}$.

(m^2 n^3)^4 (m^5 n^2) = m^{2 \cdot 4 + 5} n^{3 \cdot 4 + 2} = m^{13} n^{16}

In this example, we raised the product of $m^2$ and $n^3$ to the power of 4, resulting in $m^{8} n^{12}$, and then multiplied it by $m^5 n^2$, resulting in $m^{13} n^{16}$.

Conclusion

In this article, we have explored the concept of simplifying expressions that involve products and powers. We have used the power to a power rule and the product rule to simplify expressions and provide examples to illustrate the concepts. By following these rules, we can simplify complex expressions and arrive at the desired result.

Final Answer

The final answer is $\boxed{m^7 n^6}$.

References

• Algebraic Expressions
• Power to a Power Rule
• Product Rule

Additional Resources

• Algebraic Expressions Tutorial
• Power to a Power Rule Tutorial
• Product Rule Tutorial
  Frequently Asked Questions (FAQs) =====================================

Q: What is the power to a power rule?

A: The power to a power rule is a mathematical rule that states that when a power is raised to another power, the exponents are multiplied. For example, $(a^m)^n = a^{m \cdot n}$.

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

A: To apply the power to a power rule, simply multiply the exponents of the two powers. For example, $(m^2 n^3)^4 = m^{2 \cdot 4} n^{3 \cdot 4} = m^8 n^{12}$.

Q: What is the product rule?

A: The product rule is a mathematical rule that states that when two or more powers are multiplied together, the exponents are added. For example, $a^m \cdot a^n = a^{m + n}$.

Q: How do I apply the product rule?

A: To apply the product rule, simply add the exponents of the two powers. For example, $m^2 n^3 \cdot m^5 n^2 = m^{2 + 5} n^{3 + 2} = m^7 n^5$.

Q: Can I apply both the power to a power rule and the product rule to the same expression?

A: Yes, you can apply both the power to a power rule and the product rule to the same expression. For example, $(m^2 n^3)^4 (m^5 n^2) = m^{2 \cdot 4 + 5} n^{3 \cdot 4 + 2} = m^{13} n^{16}$.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you can use the rule that $a^{-m} = \frac{1}{a^m}$. For example, $m^{-3} = \frac{1}{m^3}$.

Q: Can I simplify expressions with fractional exponents?

A: Yes, you can simplify expressions with fractional exponents. For example, $m^{\frac{1}{2}} = \sqrt{m}$.

Q: How do I simplify expressions with multiple bases?

A: To simplify expressions with multiple bases, you can use the rule that $a^m \cdot b^n = (ab)^{m + n}$. For example, $m^2 n^3 \cdot p^5 q^2 = (mnp)^{2 + 5} (pq)^{3 + 2} = (mnp)^7 (pq)^5$.

Q: Can I simplify expressions with variables in the exponent?

A: Yes, you can simplify expressions with variables in the exponent. For example, $m^{2x} n^{3x} = (m^2 n^3)^x$.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying expressions with products and powers. We have covered topics such as the power to a power rule, the product rule, negative exponents, fractional exponents, and multiple bases. By following these rules and examples, you can simplify complex expressions and arrive at the desired result.

Final Answer

The final answer is $\boxed{m^7 n^6}$.

References

• Algebraic Expressions
• Power to a Power Rule
• Product Rule

Additional Resources

• Algebraic Expressions Tutorial
• Power to a Power Rule Tutorial
• Product Rule Tutorial