Use Properties Of Exponents To Simplify The Expression. Assume That Any Variables In Denominators Are Not Equal To Zero.$\[ \frac{84 A^7 B^4}{-14 A^9 B^{-7}} = \square \\](Simplify Your Answer. Use Exponential Notation With Positive Exponents.)

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Introduction

When dealing with algebraic expressions, simplifying them is an essential step to make them easier to work with. One of the most powerful tools for simplifying expressions is the use of properties of exponents. In this article, we will explore how to use properties of exponents to simplify a given expression.

Understanding Exponents

Before we dive into simplifying expressions, let's quickly review what exponents are. An exponent is a small number that is placed above and to the right of a number or a variable. It tells us how many times to multiply the base number or variable by itself. For example, in the expression a3a^3, the exponent 3 tells us to multiply the base aa by itself three times: a3=aâ‹…aâ‹…aa^3 = a \cdot a \cdot a.

Properties of Exponents

There are several properties of exponents that we will use to simplify expressions. These properties are:

  • Product of Powers Property: When multiplying two powers with the same base, we add the exponents. For example, a2â‹…a3=a2+3=a5a^2 \cdot a^3 = a^{2+3} = a^5.
  • Power of a Power Property: When raising a power to another power, we multiply the exponents. For example, (a2)3=a2â‹…3=a6(a^2)^3 = a^{2 \cdot 3} = a^6.
  • Quotient of Powers Property: When dividing two powers with the same base, we subtract the exponents. For example, a2a3=a2−3=a−1\frac{a^2}{a^3} = a^{2-3} = a^{-1}.
  • Zero Exponent Property: Any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

Simplifying the Expression

Now that we have reviewed the properties of exponents, let's apply them to simplify the given expression:

{ \frac{84 a^7 b^4}{-14 a^9 b^{-7}} = \square \}

To simplify this expression, we will use the quotient of powers property, which states that when dividing two powers with the same base, we subtract the exponents.

First, let's simplify the coefficients:

{ \frac{84}{-14} = -6 \}

Next, let's simplify the exponents:

{ \frac{a^7}{a^9} = a^{7-9} = a^{-2} \}

{ \frac{b^4}{b^{-7}} = b^{4-(-7)} = b^{4+7} = b^{11} \}

Now, let's substitute the simplified coefficients and exponents back into the original expression:

{ \frac{-6 a^{-2} b^{11}}{1} = -6 a^{-2} b^{11} \}

Finally, let's simplify the expression by applying the zero exponent property, which states that any non-zero number raised to the power of zero is equal to 1. In this case, we have a−2a^{-2}, which can be rewritten as 1a2\frac{1}{a^2}.

{ -6 a^{-2} b^{11} = -6 \cdot \frac{1}{a^2} \cdot b^{11} = -6 \frac{b^{11}}{a^2} \}

Conclusion

In this article, we have learned how to use properties of exponents to simplify a given expression. We have reviewed the product of powers property, power of a power property, quotient of powers property, and zero exponent property. We have then applied these properties to simplify the expression 84a7b4−14a9b−7\frac{84 a^7 b^4}{-14 a^9 b^{-7}}. The simplified expression is −6b11a2-6 \frac{b^{11}}{a^2}.

Practice Problems

  1. Simplify the expression 16a3b2−8a5b−4\frac{16 a^3 b^2}{-8 a^5 b^{-4}} using properties of exponents.
  2. Simplify the expression 25a4b35a2b−1\frac{25 a^4 b^3}{5 a^2 b^{-1}} using properties of exponents.
  3. Simplify the expression 36a2b5−9a4b−2\frac{36 a^2 b^5}{-9 a^4 b^{-2}} using properties of exponents.

Answer Key

  1. 2a−2b61=2a−2b6=2b6a2\frac{2 a^{-2} b^6}{1} = 2 a^{-2} b^6 = \frac{2 b^6}{a^2}
  2. 5a2b41=5a2b4\frac{5 a^2 b^4}{1} = 5 a^2 b^4
  3. 4a−2b71=4a−2b7=4b7a2\frac{4 a^{-2} b^7}{1} = 4 a^{-2} b^7 = \frac{4 b^7}{a^2}
    Frequently Asked Questions: Simplifying Expressions with Exponents ====================================================================

Q: What are exponents and why are they important in algebra?

A: Exponents are small numbers that are placed above and to the right of a number or a variable. They tell us how many times to multiply the base number or variable by itself. Exponents are important in algebra because they allow us to simplify complex expressions and make them easier to work with.

Q: What are the properties of exponents?

A: There are several properties of exponents that we use to simplify expressions. These properties are:

  • Product of Powers Property: When multiplying two powers with the same base, we add the exponents. For example, a2â‹…a3=a2+3=a5a^2 \cdot a^3 = a^{2+3} = a^5.
  • Power of a Power Property: When raising a power to another power, we multiply the exponents. For example, (a2)3=a2â‹…3=a6(a^2)^3 = a^{2 \cdot 3} = a^6.
  • Quotient of Powers Property: When dividing two powers with the same base, we subtract the exponents. For example, a2a3=a2−3=a−1\frac{a^2}{a^3} = a^{2-3} = a^{-1}.
  • Zero Exponent Property: Any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to follow these steps:

  1. Simplify the coefficients (numbers) in the expression.
  2. Simplify the exponents (powers) in the expression using the properties of exponents.
  3. Substitute the simplified coefficients and exponents back into the original expression.
  4. Simplify the expression further if possible.

Q: What is the quotient of powers property and how do I use it?

A: The quotient of powers property states that when dividing two powers with the same base, we subtract the exponents. For example, a2a3=a2−3=a−1\frac{a^2}{a^3} = a^{2-3} = a^{-1}. To use this property, you need to identify the base and the exponents in the expression, and then subtract the exponents.

Q: What is the zero exponent property and how do I use it?

A: The zero exponent property states that any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1. To use this property, you need to identify the base and the exponent in the expression, and then simplify the expression by replacing the exponent with 1.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents. To do this, you need to use the quotient of powers property to rewrite the negative exponent as a positive exponent. For example, a−2=1a2a^{-2} = \frac{1}{a^2}.

Q: Can I simplify an expression with fractional exponents?

A: Yes, you can simplify an expression with fractional exponents. To do this, you need to use the power of a power property to rewrite the fractional exponent as a product of powers. For example, a12=aa^{\frac{1}{2}} = \sqrt{a}.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying expressions with exponents. We have reviewed the properties of exponents, including the product of powers property, power of a power property, quotient of powers property, and zero exponent property. We have also provided examples of how to use these properties to simplify expressions with exponents.