Simplify:$\[ \frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}} \\]Write Your Answer Using Only A Positive Exponent. Assume That The Variable Represents A Positive Real Number.

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Introduction

When dealing with exponents, it's essential to understand the rules of exponentiation to simplify expressions. In this case, we're given the expression $\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}}$, and we're asked to simplify it using only a positive exponent. This means we need to apply the rules of exponentiation to combine the exponents and eliminate any negative exponents.

The Rule of Exponent Division

To simplify the given expression, we'll use the rule of exponent division, which states that when dividing two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base, and $m$ and $n$ are the exponents.

Applying the Rule of Exponent Division

Now, let's apply the rule of exponent division to the given expression:

$$\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}} = w^{\frac{4}{7} - \frac{2}{3}}$$

Simplifying the Exponent

To simplify the exponent, we need to find a common denominator. The least common multiple (LCM) of 7 and 3 is 21. So, we can rewrite the exponents with a common denominator of 21:

$$\frac{4}{7} = \frac{4 \times 3}{7 \times 3} = \frac{12}{21}$$

$$\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}$$

Now, we can subtract the exponents:

$$w^{\frac{4}{7} - \frac{2}{3}} = w^{\frac{12}{21} - \frac{14}{21}} = w^{-\frac{2}{21}}$$

Eliminating Negative Exponents

Since we're asked to simplify the expression using only a positive exponent, we need to eliminate the negative exponent. To do this, we'll use the rule of negative exponents, which states that:

$$a^{-n} = \frac{1}{a^n}$$

So, we can rewrite the expression as:

$$w^{-\frac{2}{21}} = \frac{1}{w^{\frac{2}{21}}}$$

Simplifying the Expression

Now, we can simplify the expression by rewriting it with a positive exponent:

$$\frac{1}{w^{\frac{2}{21}}} = w^{-\frac{2}{21}} \cdot w^{\frac{2}{21}} = w^{\frac{2}{21} - \frac{2}{21}} = w^0$$

Conclusion

Using the rules of exponentiation, we've simplified the expression $\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}}$ to $w^0$. This is a fundamental property of exponents, which states that any non-zero number raised to the power of 0 is equal to 1.

Final Answer

The final answer is $\boxed{1}$.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to remember the following common mistakes to avoid:

• Not using the rule of exponent division when dividing two powers with the same base.
• Not finding a common denominator when simplifying the exponent.
• Not eliminating negative exponents using the rule of negative exponents.
• Not simplifying the expression to a positive exponent.

By avoiding these common mistakes, you'll be able to simplify expressions with exponents accurately and efficiently.

Practice Problems

To practice simplifying expressions with exponents, try the following problems:

• Simplify $\frac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}}$
• Simplify $\frac{y^{\frac{2}{5}}}{y^{\frac{3}{10}}}$
• Simplify $\frac{z^{\frac{4}{9}}}{z^{\frac{2}{3}}}$

Remember to use the rules of exponentiation and to simplify the expression to a positive exponent.

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Introduction

In our previous article, we simplified the expression $\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}}$ using the rules of exponentiation. In this article, we'll answer some frequently asked questions (FAQs) related to simplifying expressions with exponents.

Q&A

Q: What is the rule of exponent division?

A: The rule of exponent division states that when dividing two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base, and $m$ and $n$ are the exponents.

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

A: To simplify an expression with a negative exponent, we'll use the rule of negative exponents, which states that:

$$a^{-n} = \frac{1}{a^n}$$

So, we can rewrite the expression as:

$$a^{-n} = \frac{1}{a^n}$$

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

A: A positive exponent represents a power of the base, while a negative exponent represents the reciprocal of the base. For example:

$$a^m = a \cdot a \cdot a \cdot ... \cdot a \quad (m \text{ times})$$

$$a^{-n} = \frac{1}{a \cdot a \cdot a \cdot ... \cdot a} \quad (n \text{ times})$$

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

A: To simplify an expression with a fraction exponent, we'll use the rule of exponent division, which states that when dividing two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

So, we can rewrite the expression as:

$$a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$$

Q: What is the difference between a rational exponent and a radical exponent?

A: A rational exponent is an exponent that is a fraction, while a radical exponent is an exponent that is a root. For example:

$$a^{\frac{1}{2}} = \sqrt{a}$$

$$a^{\frac{2}{3}} = \sqrt[3]{a^2}$$

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

A: To simplify an expression with a radical exponent, we'll use the rule of radical exponents, which states that:

$$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

So, we can rewrite the expression as:

$$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

Q: What is the final answer to the expression $\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}}$?

A: The final answer to the expression $\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}}$ is $w^0$, which is equal to 1.

Conclusion

In this article, we've answered some frequently asked questions (FAQs) related to simplifying expressions with exponents. We've covered topics such as the rule of exponent division, simplifying expressions with negative exponents, and simplifying expressions with fraction exponents. We've also discussed the difference between rational exponents and radical exponents, and how to simplify expressions with radical exponents.

Final Answer

The final answer to the expression $\frac{w^{\frac{4}{7}}}{w^{\frac{2}{3}}}$ is $\boxed{1}$.

Practice Problems

To practice simplifying expressions with exponents, try the following problems:

• Simplify $\frac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}}$
• Simplify $\frac{y^{\frac{2}{5}}}{y^{\frac{3}{10}}}$
• Simplify $\frac{z^{\frac{4}{9}}}{z^{\frac{2}{3}}}$
• Simplify $\sqrt[3]{x^2}$
• Simplify $\sqrt[4]{y^3}$