Apply The Exponent Properties To Simplify The Given Expression:$ \frac{12 Y^{10}}{16 Y^{13} X^{-2}} }$Choose The Correct Simplified Form A. ${$4 Y^3 X^2$ $B. { \frac{3 X^2}{4 Y^3}$}$C. { \frac{3 Y^3}{4 X^2}$}$D.

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Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the exponent properties and apply them to simplify a given expression. We will also discuss the importance of simplifying exponential expressions and provide a step-by-step guide on how to do it.

What are Exponent Properties?

Exponent properties are rules that help us simplify exponential expressions. There are several exponent properties, including:

  • Product of Powers Property: This property states that when we multiply two or more exponential expressions with the same base, we can add their exponents.
  • Power of a Power Property: This property states that when we raise an exponential expression to a power, we can multiply the exponents.
  • Quotient of Powers Property: This property states that when we divide two or more exponential expressions with the same base, we can subtract their exponents.
  • Zero Exponent Property: This property states that any non-zero number raised to the power of zero is equal to 1.

Simplifying the Given Expression

Now that we have discussed the exponent properties, let's apply them to simplify the given expression:

{ \frac{12 y^{10}}{16 y^{13} x^{-2}} \}

To simplify this expression, we need to apply the quotient of powers property, which states that when we divide two or more exponential expressions with the same base, we can subtract their exponents.

Step 1: Simplify the Numerator and Denominator

First, let's simplify the numerator and denominator separately.

Numerator: 12y1012 y^{10} Denominator: 16y13x−216 y^{13} x^{-2}

We can simplify the numerator by dividing 12 by 4, which gives us 3. We can also simplify the denominator by dividing 16 by 4, which gives us 4.

Numerator: 3y103 y^{10} Denominator: 4y13x−24 y^{13} x^{-2}

Step 2: Apply the Quotient of Powers Property

Now that we have simplified the numerator and denominator, we can apply the quotient of powers property to simplify the expression.

{ \frac{3 y^{10}}{4 y^{13} x^{-2}} = \frac{3}{4} \cdot \frac{y^{10}}{y^{13}} \cdot \frac{1}{x^{-2}} \}

We can simplify the expression by subtracting the exponents of the numerator and denominator, which gives us:

{ \frac{3}{4} \cdot y^{10-13} \cdot x^2 \}

Step 3: Simplify the Expression

Now that we have applied the quotient of powers property, we can simplify the expression further.

{ \frac{3}{4} \cdot y^{-3} \cdot x^2 \}

We can simplify the expression by multiplying the coefficients and adding the exponents, which gives us:

{ \frac{3 x^2}{4 y^3} \}

Conclusion

In this article, we have discussed the exponent properties and applied them to simplify a given expression. We have also provided a step-by-step guide on how to simplify exponential expressions. By following these steps, you can simplify any exponential expression and make it easier to work with.

Answer

The correct simplified form of the given expression is:

{ \frac{3 x^2}{4 y^3} \}

This is option B.

Importance of Simplifying Exponential Expressions

Simplifying exponential expressions is an essential skill for any math enthusiast. It helps us to:

  • Understand the concept of exponents: Simplifying exponential expressions helps us to understand the concept of exponents and how they work.
  • Solve equations and inequalities: Simplifying exponential expressions helps us to solve equations and inequalities that involve exponents.
  • Work with complex expressions: Simplifying exponential expressions helps us to work with complex expressions that involve exponents.
  • Make calculations easier: Simplifying exponential expressions helps us to make calculations easier and faster.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid:

  • Not applying the quotient of powers property: Failing to apply the quotient of powers property can lead to incorrect simplifications.
  • Not simplifying the numerator and denominator separately: Failing to simplify the numerator and denominator separately can lead to incorrect simplifications.
  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect simplifications.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

  1. Simplify the expression: ${ \frac{2 x^5}{3 x^2} }$
  2. Simplify the expression: ${ \frac{4 y^3}{2 y^2} }$
  3. Simplify the expression: ${ \frac{3 z^4}{2 z^3} }$

Conclusion

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two or more exponential expressions with the same base, we can add their exponents. For example, amâ‹…an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the power of a power property?

A: The power of a power property states that when we raise an exponential expression to a power, we can multiply the exponents. For example, (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}.

Q: What is the quotient of powers property?

A: The quotient of powers property states that when we divide two or more exponential expressions with the same base, we can subtract their exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: What is the zero exponent property?

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.

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

A: To simplify an exponential expression with a negative exponent, we can use the quotient of powers property. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}. If the exponent is negative, we can rewrite it as a positive exponent by changing the sign of the exponent. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

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

A: To simplify an exponential expression with a fractional exponent, we can use the power of a power property. For example, (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. If the exponent is a fraction, we can rewrite it as a product of two exponents. For example, amn=(am)1na^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}.

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

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

  • Not applying the quotient of powers property
  • Not simplifying the numerator and denominator separately
  • Not following the order of operations
  • Not using the correct exponent properties

Q: How can I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working through examples and exercises. You can also try simplifying expressions with different bases and exponents. Additionally, you can use online resources and tools to help you practice and improve your skills.

Q: What are some real-world applications of simplifying exponential expressions?

A: Simplifying exponential expressions has many real-world applications, including:

  • Calculating interest rates and investments
  • Modeling population growth and decay
  • Analyzing data and statistics
  • Solving problems in physics and engineering

Q: Can I use a calculator to simplify exponential expressions?

A: Yes, you can use a calculator to simplify exponential expressions. However, it's always a good idea to double-check your work and make sure that you understand the underlying math. Additionally, using a calculator can help you to check your work and ensure that you are getting the correct answer.

Q: How can I use simplifying exponential expressions in my daily life?

A: Simplifying exponential expressions can be useful in many areas of your daily life, including:

  • Calculating interest rates and investments
  • Modeling population growth and decay
  • Analyzing data and statistics
  • Solving problems in physics and engineering

By understanding and applying the exponent properties, you can simplify exponential expressions and make them easier to work with. Whether you're a student, a professional, or just someone who wants to improve their math skills, simplifying exponential expressions is an essential skill to have.