Simplify The Expression:$\frac{u^{\frac{7}{9}}}{u^{\frac{1}{3}}}$Write Your Answer Using Only A Positive Exponent. Assume That The Variable Represents A Positive Real Number.

Introduction

In algebra, simplifying exponential expressions is a crucial skill that helps us solve equations and manipulate mathematical expressions. In this article, we will focus on simplifying the expression $\frac{u^{\frac{7}{9}}}{u^{\frac{1}{3}}}$, where $u$ represents a positive real number. We will use the properties of exponents to rewrite the expression in a simpler form.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, $u^3$ means $u$ multiplied by itself three times: $u \cdot u \cdot u$.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the expression $\frac{u^{\frac{7}{9}}}{u^{\frac{1}{3}}}$. To do this, we will use the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a positive real number and $m$ and $n$ are exponents.

Using this property, we can rewrite the expression as:

$$\frac{u^{\frac{7}{9}}}{u^{\frac{1}{3}}} = u^{\frac{7}{9} - \frac{1}{3}}$$

Simplifying the Exponent

Now that we have rewritten the expression, let's simplify the exponent. To do this, we will use the property of exponents that states $a^m - a^n = a^{m-n}$, where $a$ is a positive real number and $m$ and $n$ are exponents.

Using this property, we can rewrite the exponent as:

$$\frac{7}{9} - \frac{1}{3} = \frac{7}{9} - \frac{3}{9} = \frac{4}{9}$$

Final Answer

Now that we have simplified the exponent, we can rewrite the expression as:

$$u^{\frac{4}{9}}$$

This is the final answer to the problem. We have successfully simplified the expression $\frac{u^{\frac{7}{9}}}{u^{\frac{1}{3}}}$ using the properties of exponents.

Conclusion

Simplifying exponential expressions is an important skill in algebra that helps us solve equations and manipulate mathematical expressions. In this article, we used the properties of exponents to simplify the expression $\frac{u^{\frac{7}{9}}}{u^{\frac{1}{3}}}$. We rewrote the expression using the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$, and then simplified the exponent using the property of exponents that states $a^m - a^n = a^{m-n}$. The final answer to the problem is $u^{\frac{4}{9}}$.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. Here are a few:

• Not using the correct property of exponents: Make sure to use the correct property of exponents when simplifying the expression. In this case, we used the property $\frac{a^m}{a^n} = a^{m-n}$.
• Not simplifying the exponent: Make sure to simplify the exponent using the property of exponents that states $a^m - a^n = a^{m-n}$.
• Not checking the final answer: Make sure to check the final answer to ensure that it is correct.

Practice Problems

Here are a few practice problems to help you practice simplifying exponential expressions:

• $\frac{u^{\frac{2}{5}}}{u^{\frac{1}{5}}}$
• $\frac{u^{\frac{3}{4}}}{u^{\frac{1}{4}}}$
• $\frac{u^{\frac{5}{6}}}{u^{\frac{1}{6}}}$

Answer Key

Here is the answer key for the practice problems:

• $\frac{u^{\frac{2}{5}}}{u^{\frac{1}{5}}} = u^{\frac{1}{5}}$
• $\frac{u^{\frac{3}{4}}}{u^{\frac{1}{4}}} = u^{\frac{2}{4}} = u^{\frac{1}{2}}$
• $\frac{u^{\frac{5}{6}}}{u^{\frac{1}{6}}} = u^{\frac{4}{6}} = u^{\frac{2}{3}}$

Conclusion

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Introduction

In our previous article, we discussed how to simplify exponential expressions using the properties of exponents. In this article, we will provide a Q&A guide to help you understand the concepts and apply them to different problems.

Q: What is the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$?

A: This property states that when we divide two exponential expressions with the same base, we can subtract the exponents. For example, $\frac{u^3}{u^2} = u^{3-2} = u^1$.

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 property of exponents that states $a^{-m} = \frac{1}{a^m}$. For example, $u^{-2} = \frac{1}{u^2}$.

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This property states that when we multiply two exponential expressions with the same base, we can add the exponents. For example, $u^2 \cdot u^3 = u^{2+3} = u^5$.

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 property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, $u^{\frac{1}{2}} = \sqrt{u}$.

Q: What is the property of exponents that states $(a^m)^n = a^{m \cdot n}$?

A: This property states that when we raise an exponential expression to a power, we can multiply the exponents. For example, $(u^2)^3 = u^{2 \cdot 3} = u^6$.

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

A: To simplify an exponential expression with a zero exponent, we can use the property of exponents that states $a^0 = 1$. For example, $u^0 = 1$.

Q: What is the property of exponents that states $\frac{1}{a^m} = a^{-m}$?

A: This property states that when we divide 1 by an exponential expression, we can change the sign of the exponent. For example, $\frac{1}{u^2} = u^{-2}$.

Q: How do I simplify an exponential expression with a variable base?

A: To simplify an exponential expression with a variable base, we can use the properties of exponents that we have discussed. For example, $\frac{u^3}{u^2} = u^{3-2} = u^1$.

Conclusion

Simplifying exponential expressions is an important skill in algebra that helps us solve equations and manipulate mathematical expressions. In this article, we provided a Q&A guide to help you understand the concepts and apply them to different problems. We discussed the properties of exponents, including the property that states $\frac{a^m}{a^n} = a^{m-n}$, and how to simplify exponential expressions with negative, fractional, and zero exponents. We also discussed how to simplify exponential expressions with variable bases. By following these steps and practicing with different problems, you will become more confident in your ability to simplify exponential expressions.

Practice Problems

Here are a few practice problems to help you practice simplifying exponential expressions:

• $\frac{u^{\frac{2}{3}}}{u^{\frac{1}{3}}}$
• $\frac{u^{\frac{3}{4}}}{u^{\frac{1}{4}}}$
• $\frac{u^{\frac{5}{6}}}{u^{\frac{1}{6}}}$
• $(u^2)^3$
• $\frac{1}{u^2}$
• $u^0$
• $\frac{u^3}{u^2}$

Answer Key

Here is the answer key for the practice problems:

• $\frac{u^{\frac{2}{3}}}{u^{\frac{1}{3}}} = u^{\frac{1}{3}}$
• $\frac{u^{\frac{3}{4}}}{u^{\frac{1}{4}}} = u^{\frac{2}{4}} = u^{\frac{1}{2}}$
• $\frac{u^{\frac{5}{6}}}{u^{\frac{1}{6}}} = u^{\frac{4}{6}} = u^{\frac{2}{3}}$
• $(u^2)^3 = u^{2 \cdot 3} = u^6$
• $\frac{1}{u^2} = u^{-2}$
• $u^0 = 1$
• $\frac{u^3}{u^2} = u^{3-2} = u^1$