Simplify The Expression:$\[ \frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}} \\]Write Your Answer Using Only A Positive Exponent. Assume That The Variable Represents A Positive Number.

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Introduction

When dealing with exponents, it's essential to understand the rules of exponentiation to simplify expressions. In this article, we will focus on simplifying the given expression $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$ using the rules of exponentiation. We will assume that the variable represents a positive number.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ can be written as $x \cdot x \cdot x$. When we have a fraction as an exponent, such as $\frac{2}{7}$, it means that we are raising the base to the power of $\frac{2}{7}$.

Simplifying the Expression

To simplify the given expression, we can use the rule of dividing exponents with the same base, which states that $\frac{x^a}{x^b} = x^{a-b}$. In this case, we have:

$$\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}} = x^{\frac{2}{7} - \frac{3}{5}}$$

Calculating the Exponent

To calculate the exponent, we need to find a common denominator. The least common multiple of 7 and 5 is 35. We can rewrite the exponents as:

$$\frac{2}{7} = \frac{2 \cdot 5}{7 \cdot 5} = \frac{10}{35}$$

$$\frac{3}{5} = \frac{3 \cdot 7}{5 \cdot 7} = \frac{21}{35}$$

Now, we can subtract the exponents:

$$\frac{10}{35} - \frac{21}{35} = \frac{-11}{35}$$

Simplifying the Result

Since we are assuming that the variable represents a positive number, we can simplify the result by taking the reciprocal of the exponent:

$$x^{\frac{-11}{35}} = \frac{1}{x^{\frac{11}{35}}}$$

Conclusion

In this article, we simplified the expression $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$ using the rules of exponentiation. We assumed that the variable represents a positive number and used the rule of dividing exponents with the same base to simplify the expression. The final result is $\frac{1}{x^{\frac{11}{35}}}$.

Frequently Asked Questions

• What is the rule of dividing exponents with the same base? The rule of dividing exponents with the same base states that $\frac{x^a}{x^b} = x^{a-b}$.
• How do I calculate the exponent when dividing exponents with the same base? To calculate the exponent, you need to find a common denominator and subtract the exponents.
• What is the final result of simplifying the expression $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$? The final result is $\frac{1}{x^{\frac{11}{35}}}$.

Additional Resources

• Exponent Rules: A comprehensive guide to exponent rules, including the rule of dividing exponents with the same base.
• Simplifying Expressions: A tutorial on simplifying expressions using the rules of exponentiation.
• Mathematics: A comprehensive guide to mathematics, including topics such as algebra, geometry, and calculus.
  

Introduction

In our previous article, we simplified the expression $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$ using the rules of exponentiation. In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q&A

Q: What is the rule of dividing exponents with the same base?

A: The rule of dividing exponents with the same base states that $\frac{x^a}{x^b} = x^{a-b}$.

Q: How do I calculate the exponent when dividing exponents with the same base?

A: To calculate the exponent, you need to find a common denominator and subtract the exponents. For example, if you have $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$, you can find a common denominator of 35 and rewrite the exponents as $\frac{10}{35}$ and $\frac{21}{35}$. Then, you can subtract the exponents to get $\frac{-11}{35}$.

Q: What is the final result of simplifying the expression $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$?

A: The final result is $\frac{1}{x^{\frac{11}{35}}}$.

Q: Can I simplify expressions with exponents that have different bases?

A: Yes, you can simplify expressions with exponents that have different bases by using the rule of multiplying exponents with the same base. For example, if you have $\frac{x^a}{y^b}$, you can rewrite it as $x^a \cdot y^{-b}$.

Q: How do I simplify expressions with exponents that have negative exponents?

A: To simplify expressions with exponents that have negative exponents, you can use the rule of negative exponents, which states that $x^{-a} = \frac{1}{x^a}$.

Q: Can I simplify expressions with exponents that have fractional exponents?

A: Yes, you can simplify expressions with exponents that have fractional exponents by using the rule of fractional exponents, which states that $x^{\frac{a}{b}} = \sqrt[b]{x^a}$.

Examples

Example 1: Simplifying an Expression with Exponents

Simplify the expression $\frac{x^{\frac{3}{4}}}{x^{\frac{2}{3}}}$.

Solution:

To simplify the expression, we can use the rule of dividing exponents with the same base:

$$\frac{x^{\frac{3}{4}}}{x^{\frac{2}{3}}} = x^{\frac{3}{4} - \frac{2}{3}}$$

To calculate the exponent, we need to find a common denominator. The least common multiple of 4 and 3 is 12. We can rewrite the exponents as:

$$\frac{3}{4} = \frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12}$$

$$\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}$$

Now, we can subtract the exponents:

$$\frac{9}{12} - \frac{8}{12} = \frac{1}{12}$$

So, the simplified expression is $x^{\frac{1}{12}}$.

Example 2: Simplifying an Expression with Negative Exponents

Simplify the expression $\frac{1}{x^{\frac{2}{3}}}$.

Solution:

To simplify the expression, we can use the rule of negative exponents, which states that $x^{-a} = \frac{1}{x^a}$.

So, we can rewrite the expression as:

$$\frac{1}{x^{\frac{2}{3}}} = x^{-\frac{2}{3}}$$

Now, we can use the rule of negative exponents to simplify the expression:

$$x^{-\frac{2}{3}} = \frac{1}{x^{\frac{2}{3}}}$$

So, the simplified expression is $\frac{1}{x^{\frac{2}{3}}}$.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with exponents. We covered topics such as the rule of dividing exponents with the same base, calculating exponents, and simplifying expressions with negative exponents and fractional exponents. We also provided examples to illustrate the concepts.

Frequently Asked Questions

• What is the rule of dividing exponents with the same base? The rule of dividing exponents with the same base states that $\frac{x^a}{x^b} = x^{a-b}$.
• How do I calculate the exponent when dividing exponents with the same base? To calculate the exponent, you need to find a common denominator and subtract the exponents.
• What is the final result of simplifying the expression $\frac{x^{\frac{2}{7}}}{x^{\frac{3}{5}}}$? The final result is $\frac{1}{x^{\frac{11}{35}}}$.

Additional Resources

• Exponent Rules: A comprehensive guide to exponent rules, including the rule of dividing exponents with the same base.
• Simplifying Expressions: A tutorial on simplifying expressions using the rules of exponentiation.
• Mathematics: A comprehensive guide to mathematics, including topics such as algebra, geometry, and calculus.