Simplify The Following Expressions, Expressing The Answers Using Positive Exponents:1. { \frac{-2a 2b 4}{4ab^{-8}}, A \neq 0, B \neq 0$}$2. { \frac{-5w 4y {-2}}{-15w {-6}y 2}, W \neq 0, Y \neq 0$}$Express The Simplified Answers

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Introduction

Algebraic expressions with negative exponents can be simplified using the properties of exponents. In this article, we will simplify two given expressions, expressing the answers using positive exponents. We will use the properties of exponents, such as the product of powers property and the quotient of powers property, to simplify the expressions.

Simplifying the First Expression

The first expression is given by:

βˆ’2a2b44abβˆ’8,aβ‰ 0,bβ‰ 0\frac{-2a^2b^4}{4ab^{-8}}, a \neq 0, b \neq 0

To simplify this expression, we will use the quotient of powers property, which states that:

aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}

We can rewrite the expression as:

βˆ’2a2b44abβˆ’8=βˆ’2a2b44aβ‹…bβˆ’8\frac{-2a^2b^4}{4ab^{-8}} = \frac{-2a^2b^4}{4a \cdot b^{-8}}

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

βˆ’2a2b44aβ‹…bβˆ’8=βˆ’2a2b44aβ‹…1bβˆ’8\frac{-2a^2b^4}{4a \cdot b^{-8}} = \frac{-2a^2b^4}{4a} \cdot \frac{1}{b^{-8}}

We can further simplify the expression by using the product of powers property, which states that:

amβ‹…an=am+na^m \cdot a^n = a^{m+n}

We can rewrite the expression as:

βˆ’2a2b44aβ‹…1bβˆ’8=βˆ’2a2b44aβ‹…b8\frac{-2a^2b^4}{4a} \cdot \frac{1}{b^{-8}} = \frac{-2a^2b^4}{4a} \cdot b^8

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

βˆ’2a2b44aβ‹…b8=βˆ’2a2b44aβ‹…b8=βˆ’2a2b4+84a\frac{-2a^2b^4}{4a} \cdot b^8 = \frac{-2a^2b^4}{4a} \cdot b^8 = \frac{-2a^2b^{4+8}}{4a}

We can further simplify the expression by using the quotient of powers property:

βˆ’2a2b4+84a=βˆ’2a2b124a=βˆ’2a2βˆ’1b124\frac{-2a^2b^{4+8}}{4a} = \frac{-2a^2b^{12}}{4a} = \frac{-2a^{2-1}b^{12}}{4}

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

βˆ’2a2βˆ’1b124=βˆ’2a1b124=βˆ’2ab124\frac{-2a^{2-1}b^{12}}{4} = \frac{-2a^{1}b^{12}}{4} = \frac{-2ab^{12}}{4}

We can further simplify the expression by dividing both the numerator and the denominator by 2:

βˆ’2ab124=βˆ’ab122\frac{-2ab^{12}}{4} = \frac{-ab^{12}}{2}

Therefore, the simplified expression is:

βˆ’ab122\frac{-ab^{12}}{2}

Simplifying the Second Expression

The second expression is given by:

βˆ’5w4yβˆ’2βˆ’15wβˆ’6y2,wβ‰ 0,yβ‰ 0\frac{-5w^4y^{-2}}{-15w^{-6}y^2}, w \neq 0, y \neq 0

To simplify this expression, we will use the quotient of powers property, which states that:

aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}

We can rewrite the expression as:

βˆ’5w4yβˆ’2βˆ’15wβˆ’6y2=βˆ’5w4yβˆ’2βˆ’15wβˆ’6y2\frac{-5w^4y^{-2}}{-15w^{-6}y^2} = \frac{-5w^4y^{-2}}{-15w^{-6}y^2}

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

βˆ’5w4yβˆ’2βˆ’15wβˆ’6y2=βˆ’5w4βˆ’15wβˆ’6β‹…1y2\frac{-5w^4y^{-2}}{-15w^{-6}y^2} = \frac{-5w^4}{-15w^{-6}} \cdot \frac{1}{y^2}

We can further simplify the expression by using the product of powers property, which states that:

amβ‹…an=am+na^m \cdot a^n = a^{m+n}

We can rewrite the expression as:

βˆ’5w4βˆ’15wβˆ’6β‹…1y2=βˆ’5w4βˆ’15wβˆ’6β‹…1y2=βˆ’5w4βˆ’15β‹…1wβˆ’6β‹…1y2\frac{-5w^4}{-15w^{-6}} \cdot \frac{1}{y^2} = \frac{-5w^4}{-15w^{-6}} \cdot \frac{1}{y^2} = \frac{-5w^4}{-15} \cdot \frac{1}{w^{-6}} \cdot \frac{1}{y^2}

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

βˆ’5w4βˆ’15β‹…1wβˆ’6β‹…1y2=βˆ’5w4βˆ’15β‹…w6β‹…1y2\frac{-5w^4}{-15} \cdot \frac{1}{w^{-6}} \cdot \frac{1}{y^2} = \frac{-5w^4}{-15} \cdot w^6 \cdot \frac{1}{y^2}

We can further simplify the expression by using the quotient of powers property:

βˆ’5w4βˆ’15β‹…w6β‹…1y2=βˆ’5w4+6βˆ’15β‹…1y2\frac{-5w^4}{-15} \cdot w^6 \cdot \frac{1}{y^2} = \frac{-5w^{4+6}}{-15} \cdot \frac{1}{y^2}

We can further simplify the expression by using the quotient of powers property:

βˆ’5w4+6βˆ’15β‹…1y2=βˆ’5w10βˆ’15β‹…1y2=βˆ’5w10βˆ’15β‹…1y2=βˆ’5w10βˆ’15β‹…1y2\frac{-5w^{4+6}}{-15} \cdot \frac{1}{y^2} = \frac{-5w^{10}}{-15} \cdot \frac{1}{y^2} = \frac{-5w^{10}}{-15} \cdot \frac{1}{y^2} = \frac{-5w^{10}}{-15} \cdot \frac{1}{y^2}

We can further simplify the expression by dividing both the numerator and the denominator by 5:

βˆ’5w10βˆ’15β‹…1y2=w103β‹…1y2\frac{-5w^{10}}{-15} \cdot \frac{1}{y^2} = \frac{w^{10}}{3} \cdot \frac{1}{y^2}

We can further simplify the expression by using the product of powers property:

w103β‹…1y2=w103β‹…1y2=w103y2\frac{w^{10}}{3} \cdot \frac{1}{y^2} = \frac{w^{10}}{3} \cdot \frac{1}{y^2} = \frac{w^{10}}{3y^2}

Therefore, the simplified expression is:

w103y2\frac{w^{10}}{3y^2}

Conclusion

Introduction

In our previous article, we simplified two algebraic expressions with negative exponents, expressing the answers using positive exponents. In this article, we will answer some common questions related to simplifying algebraic expressions with negative exponents.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example, a2a^2 is a positive exponent, while aβˆ’2a^{-2} is a negative exponent.

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

A: To simplify an expression with a negative exponent, you can use the quotient of powers property, which states that:

aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}

You can also use the product of powers property, which states that:

amβ‹…an=am+na^m \cdot a^n = a^{m+n}

Q: What is the quotient of powers property?

A: The quotient of powers property states that:

aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}

This means that when you divide two powers with the same base, you can subtract the exponents.

Q: What is the product of powers property?

A: The product of powers property states that:

amβ‹…an=am+na^m \cdot a^n = a^{m+n}

This means that when you multiply two powers with the same base, you can add the exponents.

Q: How do I simplify an expression with a negative exponent and a positive exponent?

A: To simplify an expression with a negative exponent and a positive exponent, you can use the quotient of powers property and the product of powers property. For example, if you have the expression a2aβˆ’3\frac{a^2}{a^{-3}}, you can simplify it as follows:

a2aβˆ’3=a2βˆ’(βˆ’3)=a2+3=a5\frac{a^2}{a^{-3}} = a^{2-(-3)} = a^{2+3} = a^5

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

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

  • Not using the quotient of powers property or the product of powers property
  • Not simplifying the expression correctly
  • Not checking the signs of the exponents
  • Not using the correct order of operations

Q: How do I check my work when simplifying expressions with negative exponents?

A: To check your work when simplifying expressions with negative exponents, you can:

  • Use a calculator to check the expression
  • Simplify the expression using a different method
  • Check the signs of the exponents
  • Check the order of operations

Conclusion

In this article, we answered some common questions related to simplifying algebraic expressions with negative exponents. We discussed the quotient of powers property and the product of powers property, and we provided examples of how to simplify expressions with negative exponents. We also discussed some common mistakes to avoid and how to check your work when simplifying expressions with negative exponents.