Simplify Using The Properties Of Exponents. ( 3 X 2 Y − 5 Z 6 ) 4 \left(\frac{3 X^2 Y^{-5}}{z^6}\right)^4 ( Z 6 3 X 2 Y − 5 ) 4

Mar 13, 2025 by ADMIN 130 views

$Simplify using the properties of exponents.$\left(\frac{3 x^2 y^{-5}}{z^6}\right)^4$$

Understanding the Properties of Exponents

When dealing with exponents, it's essential to understand the properties that govern their behavior. The properties of exponents are a set of rules that help us simplify expressions involving exponents. In this article, we will focus on simplifying the given expression using the properties of exponents.

The Given Expression

The given expression is $\left(\frac{3 x^2 y^{-5}}{z^6}\right)^4$ . This expression involves exponents, fractions, and variables. To simplify it, we need to apply the properties of exponents.

Applying the Power Rule

The power rule states that for any variables a and b and any integer n, $(ab)^n = a^nb^n$ . This rule can be applied to the given expression by raising each factor to the power of 4.

Simplifying the Expression

Using the power rule, we can simplify the expression as follows:

$\left(\frac{3 x^2 y^{-5}}{z^6}\right)^4 = \frac{3^4 (x^2)^4 (y^{-5})^4}{(z^6)^4}$

Applying the Product Rule

The product rule states that for any variables a and b and any integers m and n, $(ab)^m = a^mb^n$ . This rule can be applied to the expression by raising each factor to the power of 4.

Simplifying the Expression Further

Using the product rule, we can simplify the expression further as follows:

$\frac{3^4 (x^2)^4 (y^{-5})^4}{(z^6)^4} = \frac{3^4 x^{2 \cdot 4} y^{-5 \cdot 4}}{z^{6 \cdot 4}}$

Applying the Quotient Rule

The quotient rule states that for any variables a and b and any integers m and n, $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ . This rule can be applied to the expression by raising each factor to the power of 4.

Simplifying the Expression Even Further

Using the quotient rule, we can simplify the expression even further as follows:

$\frac{3^4 x^{2 \cdot 4} y^{-5 \cdot 4}}{z^{6 \cdot 4}} = \frac{3^4 x^8 y^{-20}}{z^{24}}$

Applying the Negative Exponent Rule

The negative exponent rule states that for any variable a and any integer n, $a^{-n} = \frac{1}{a^n}$ . This rule can be applied to the expression by rewriting the negative exponent as a fraction.

Simplifying the Expression to Its Final Form

Using the negative exponent rule, we can simplify the expression to its final form as follows:

$\frac{3^4 x^8 y^{-20}}{z^{24}} = \frac{3^4 x^8}{y^{20} z^{24}}$

Conclusion

In this article, we simplified the given expression using the properties of exponents. We applied the power rule, product rule, quotient rule, and negative exponent rule to simplify the expression step by step. The final simplified form of the expression is $\frac{3^4 x^8}{y^{20} z^{24}}$ .

Final Answer

The final answer is $\boxed{\frac{3^4 x^8}{y^{20} z^{24}}}$ .

Frequently Asked Questions

What are the properties of exponents? The properties of exponents are a set of rules that help us simplify expressions involving exponents.
How do you simplify an expression using the properties of exponents? To simplify an expression using the properties of exponents, you need to apply the power rule, product rule, quotient rule, and negative exponent rule step by step.
What is the final simplified form of the given expression? The final simplified form of the given expression is $\frac{3^4 x^8}{y^{20} z^{24}}$ .

References

[1] Khan Academy. (n.d.). Exponents and Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7d/x2f6f7d/exponents-and-exponential-functions
[2] Math Open Reference. (n.d.). Exponents. Retrieved from https://www.mathopenref.com/exponents.html
[3] Purplemath. (n.d.). Exponents. Retrieved from https://www.purplemath.com/modules/exponent.htm

Frequently Asked Questions

Q1: What are the properties of exponents?

A1: The properties of exponents are a set of rules that help us simplify expressions involving exponents. These rules include the power rule, product rule, quotient rule, and negative exponent rule.

Q2: How do you simplify an expression using the properties of exponents?

A2: To simplify an expression using the properties of exponents, you need to apply the power rule, product rule, quotient rule, and negative exponent rule step by step. Here's a general outline:

Apply the power rule to raise each factor to the power of the exponent.
Apply the product rule to simplify the expression by combining like terms.
Apply the quotient rule to simplify the expression by dividing like terms.
Apply the negative exponent rule to rewrite negative exponents as fractions.

Q3: What is the power rule?

A3: The power rule states that for any variables a and b and any integer n, $(ab)^n = a^nb^n$ . This rule can be applied to the expression by raising each factor to the power of the exponent.

Q4: What is the product rule?

A4: The product rule states that for any variables a and b and any integers m and n, $(ab)^m = a^mb^n$ . This rule can be applied to the expression by raising each factor to the power of the exponent.

Q5: What is the quotient rule?

A5: The quotient rule states that for any variables a and b and any integers m and n, $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ . This rule can be applied to the expression by raising each factor to the power of the exponent.

Q6: What is the negative exponent rule?

A6: The negative exponent rule states that for any variable a and any integer n, $a^{-n} = \frac{1}{a^n}$ . This rule can be applied to the expression by rewriting negative exponents as fractions.

Q7: How do you simplify an expression with multiple exponents?

A7: To simplify an expression with multiple exponents, you need to apply the power rule, product rule, quotient rule, and negative exponent rule step by step. Here's a general outline:

Apply the power rule to raise each factor to the power of the exponent.
Apply the product rule to simplify the expression by combining like terms.
Apply the quotient rule to simplify the expression by dividing like terms.
Apply the negative exponent rule to rewrite negative exponents as fractions.

Q8: What is the final simplified form of the given expression?

A8: The final simplified form of the given expression is $\frac{3^4 x^8}{y^{20} z^{24}}$ .

Q9: Can you provide an example of simplifying an expression using the properties of exponents?

A9: Here's an example:

$\left(\frac{2 x^3 y^{-2}}{z^4}\right)^5$

Using the power rule, product rule, quotient rule, and negative exponent rule, we can simplify the expression as follows:

$\left(\frac{2 x^3 y^{-2}}{z^4}\right)^5 = \frac{2^5 (x^3)^5 (y^{-2})^5}{(z^4)^5} = \frac{2^5 x^{15} y^{-10}}{z^{20}} = \frac{2^5 x^{15}}{y^{10} z^{20}}$

Q10: Where can I learn more about simplifying expressions using the properties of exponents?

A10: You can learn more about simplifying expressions using the properties of exponents by visiting the following resources:

Khan Academy: Exponents and Exponential Functions
Math Open Reference: Exponents
Purplemath: Exponents

Conclusion

In this Q&A article, we covered frequently asked questions about simplifying expressions using the properties of exponents. We provided answers to common questions and provided examples to illustrate the concepts. We hope this article has been helpful in understanding the properties of exponents and how to simplify expressions using them.