Simplify The Expression Using The Properties Of Exponents. Answer Using Only Nonnegative Exponents.\[$\frac{7 M^6}{9 M^7}\$\]

Introduction

In algebra, exponents are a fundamental concept used to represent repeated multiplication of a number. The properties of exponents are essential in simplifying expressions and solving equations. In this article, we will focus on simplifying the given expression using the properties of exponents, with the goal of expressing the result using only nonnegative exponents.

Understanding the Properties of Exponents

Before we dive into simplifying the expression, let's review the properties of exponents. The properties of exponents are as follows:

• Product of Powers Property: When multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• Zero Exponent Property: Any nonzero number raised to the zero power is equal to 1. For example, $a^0 = 1$.

Simplifying the Expression

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

$\frac{7 m^6}{9 m^7}$

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

$\frac{7 m^6}{9 m^7} = \frac{7}{9} \cdot \frac{m^6}{m^7}$

Next, we will use the quotient of powers property to simplify the fraction:

$\frac{m^6}{m^7} = m^{6-7} = m^{-1}$

Now, we can rewrite the expression as:

$\frac{7 m^6}{9 m^7} = \frac{7}{9} \cdot m^{-1}$

Expressing the Result Using Only Nonnegative Exponents

Our goal is to express the result using only nonnegative exponents. To achieve this, we will use the property of exponents that states $a^{-n} = \frac{1}{a^n}$.

$\frac{7}{9} \cdot m^{-1} = \frac{7}{9} \cdot \frac{1}{m^1}$

Now, we can simplify the expression further by combining the fractions:

$\frac{7}{9} \cdot \frac{1}{m^1} = \frac{7}{9m}$

Conclusion

In this article, we simplified the given expression using the properties of exponents, with the goal of expressing the result using only nonnegative exponents. We 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 then applied these properties to simplify the expression, resulting in the final answer of $\frac{7}{9m}$.

Key Takeaways

• The properties of exponents are essential in simplifying expressions and solving equations.
• The quotient of powers property states that when dividing two powers with the same base, subtract the exponents.
• The zero exponent property states that any nonzero number raised to the zero power is equal to 1.
• To express a result using only nonnegative exponents, we can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$.

Practice Problems

1. Simplify the expression $\frac{4 x^5}{6 x^6}$ using the properties of exponents.
2. Express the result using only nonnegative exponents.
3. Simplify the expression $\frac{3 y^3}{5 y^4}$ using the properties of exponents.
4. Express the result using only nonnegative exponents.

Answer Key

1. $\frac{2}{3} \cdot \frac{1}{x^1} = \frac{2}{3x}$
2. $\frac{2}{3x}$
3. $\frac{3}{5} \cdot \frac{1}{y^1} = \frac{3}{5y}$
4. $\frac{3}{5y}$
   Simplify the Expression Using the Properties of Exponents: Q&A ===========================================================

Introduction

In our previous article, we simplified the expression $\frac{7 m^6}{9 m^7}$ using the properties of exponents, with the goal of expressing the result using only nonnegative exponents. In this article, we will provide a Q&A section to help you better understand the concepts and properties of exponents.

Q&A

Q: What are the properties of exponents?

A: The properties of exponents are as follows:

• Product of Powers Property: When multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• Zero Exponent Property: Any nonzero number raised to the zero power is equal to 1. For example, $a^0 = 1$.

Q: How do I simplify an expression using the properties of exponents?

A: To simplify an expression using the properties of exponents, follow these steps:

1. Identify the properties of exponents that apply to the expression.
2. Apply the properties to simplify the expression.
3. Use the quotient of powers property to simplify fractions.
4. Use the zero exponent property to simplify expressions with zero exponents.

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

A: A positive exponent represents a power with a positive value, while a negative exponent represents a power with a negative value. For example, $a^3$ represents a power with a positive value, while $a^{-3}$ represents a power with a negative value.

Q: How do I express a result using only nonnegative exponents?

A: To express a result using only nonnegative exponents, use the property of exponents that states $a^{-n} = \frac{1}{a^n}$. This will allow you to rewrite negative exponents as fractions with positive exponents.

Q: What are some common mistakes to avoid when simplifying expressions using the properties of exponents?

A: Some common mistakes to avoid when simplifying expressions using the properties of exponents include:

• Forgetting to apply the properties of exponents.
• Not simplifying fractions using the quotient of powers property.
• Not using the zero exponent property to simplify expressions with zero exponents.
• Not expressing results using only nonnegative exponents.

Practice Problems

1. Simplify the expression $\frac{2 x^4}{3 x^5}$ using the properties of exponents.
2. Express the result using only nonnegative exponents.
3. Simplify the expression $\frac{4 y^2}{5 y^3}$ using the properties of exponents.
4. Express the result using only nonnegative exponents.

Answer Key

1. $\frac{2}{3} \cdot \frac{1}{x^1} = \frac{2}{3x}$
2. $\frac{2}{3x}$
3. $\frac{4}{5} \cdot \frac{1}{y^1} = \frac{4}{5y}$
4. $\frac{4}{5y}$

Conclusion

In this article, we provided a Q&A section to help you better understand the concepts and properties of exponents. We covered topics such as the properties of exponents, simplifying expressions using the properties of exponents, and expressing results using only nonnegative exponents. We also provided practice problems and answer keys to help you reinforce your understanding of the concepts.