Use The Power Rule And The Power Of A Product Or Quotient Rule To Simplify The Expression.$\left(-5 A^5 B^7 C\right)^2$\left(-5 A^5 B^7 C\right)^2 = \square$ (Type Your Answer Using Exponential Notation.)

Feb 27, 2025 by ADMIN 206 views

**Simplifying Exponential Expressions Using the Power Rule and the Power of a Product or Quotient Rule**

Introduction

In algebra, simplifying exponential expressions is a crucial skill that helps us solve equations and manipulate mathematical expressions. One of the most powerful tools for simplifying exponential expressions is the power rule, which states that for any non-zero number a and integers m and n, (a^m)^n = a^(m*n). In this article, we will use the power rule and the power of a product or quotient rule to simplify the expression (-5 a^5 b^7 c)^2.

The Power Rule

The power rule is a fundamental concept in algebra that helps us simplify exponential expressions. It states that for any non-zero number a and integers m and n, (a^m)^n = a^(m*n). This means that when we raise an exponential expression to a power, we can simply multiply the exponents.

Applying the Power Rule to the Given Expression

Now, let's apply the power rule to the given expression (-5 a^5 b^7 c)^2. Using the power rule, we can simplify the expression as follows:

(-5 a^5 b^7 c)^2 = (-5)^2 * (a^5)^2 * (b^7)^2 * c^2

Simplifying the Coefficients

The first step in simplifying the expression is to simplify the coefficients. In this case, we have (-5)^2, which is equal to 25. Therefore, the expression becomes:

25 * (a^5)^2 * (b^7)^2 * c^2

Simplifying the Exponents

The next step is to simplify the exponents. Using the power rule, we can simplify the exponents as follows:

(a^5)^2 = a^(5*2) = a^10

(b^7)^2 = b^(7*2) = b^14

Therefore, the expression becomes:

25 * a^10 * b^14 * c^2

The Power of a Product or Quotient Rule

The power of a product or quotient rule states that for any non-zero numbers a and b and integers m and n, (a*b)^m = a^m * b^m and (a/b)^m = a^m / b^m. This means that when we raise a product or quotient to a power, we can simply raise each factor to that power.

Applying the Power of a Product or Quotient Rule to the Given Expression

Now, let's apply the power of a product or quotient rule to the given expression (-5 a^5 b^7 c)^2. Using the power of a product or quotient rule, we can simplify the expression as follows:

(-5 a^5 b^7 c)^2 = (-5)^2 * (a^5)^2 * (b^7)^2 * c^2

This is the same expression we obtained using the power rule. Therefore, the power of a product or quotient rule is consistent with the power rule.

Conclusion

In conclusion, we have used the power rule and the power of a product or quotient rule to simplify the expression (-5 a^5 b^7 c)^2. We have shown that the power rule is a powerful tool for simplifying exponential expressions, and that it is consistent with the power of a product or quotient rule. By applying these rules, we can simplify complex exponential expressions and solve equations with ease.

Final Answer

Q: What is the power rule in algebra?

A: The power rule is a fundamental concept in algebra that helps us simplify exponential expressions. It states that for any non-zero number a and integers m and n, (a^m)^n = a^(m*n). This means that when we raise an exponential expression to a power, we can simply multiply the exponents.

Q: How do I apply the power rule to simplify an exponential expression?

A: To apply the power rule, simply multiply the exponents of the base number. For example, if we have the expression (a^5)^2, we can simplify it as follows:

(a^5)^2 = a^(5*2) = a^10

Q: What is the power of a product or quotient rule?

A: The power of a product or quotient rule states that for any non-zero numbers a and b and integers m and n, (a*b)^m = a^m * b^m and (a/b)^m = a^m / b^m. This means that when we raise a product or quotient to a power, we can simply raise each factor to that power.

Q: How do I apply the power of a product or quotient rule to simplify an exponential expression?

A: To apply the power of a product or quotient rule, simply raise each factor to the power. For example, if we have the expression (a*b)^2, we can simplify it as follows:

(a*b)^2 = a^2 * b^2

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

Forgetting to multiply the exponents when applying the power rule
Not raising each factor to the power when applying the power of a product or quotient rule
Not simplifying the coefficients and exponents separately

Q: How do I know when to use the power rule and when to use the power of a product or quotient rule?

A: The power rule is used when we have an exponential expression raised to a power, while the power of a product or quotient rule is used when we have a product or quotient raised to a power. If you're unsure which rule to use, try simplifying the expression using both rules and see which one works.

Q: Can I use the power rule and the power of a product or quotient rule together?

A: Yes, you can use the power rule and the power of a product or quotient rule together. For example, if we have the expression ((a*b)^2)^3, we can simplify it as follows:

((a*b)^2)^3 = (a^2 * b^2)^3 = a^6 * b^6

Q: How do I check my work when simplifying exponential expressions?

A: To check your work, try plugging the simplified expression back into the original equation and see if it's true. You can also use a calculator or a computer program to check your work.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical expressions. By understanding the power rule and the power of a product or quotient rule, you can simplify complex exponential expressions and solve equations with ease. Remember to apply the power rule and the power of a product or quotient rule carefully, and don't be afraid to check your work.