Simplify The Expression:$ \frac{z {\frac{1}{3}}}{z {-\frac{1}{5}}} $Write Your Answer Using Only Positive Exponents. Assume That All Variables Are Positive Real Numbers.

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Introduction

In this article, we will simplify the given expression $\frac{z^{\frac{1}{3}}}{z^{-\frac{1}{5}}}$. We will use the properties of exponents to rewrite the expression with only positive exponents. We will also assume that all variables are positive real numbers.

Properties of Exponents

Before we start simplifying the expression, let's review some properties of exponents. The quotient rule of exponents states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a positive real number and $m$ and $n$ are integers. The product rule of exponents states that $a^m \cdot a^n = a^{m+n}$, where $a$ is a positive real number and $m$ and $n$ are integers.

Simplifying the Expression

Now, let's simplify the given expression using the quotient rule of exponents. We have $\frac{z^{\frac{1}{3}}}{z^{-\frac{1}{5}}}$. Using the quotient rule, we can rewrite this expression as $z^{\frac{1}{3} - (-\frac{1}{5})}$.

Evaluating the Exponent

Now, let's evaluate the exponent. We have $\frac{1}{3} - (-\frac{1}{5})$. To evaluate this expression, we need to get rid of the negative sign. We can do this by adding the opposite of the expression. So, we have $\frac{1}{3} + \frac{1}{5}$.

Finding a Common Denominator

To add these fractions, we need to find a common denominator. The least common multiple of $3$ and $5$ is $15$. So, we can rewrite the fractions as $\frac{5}{15}$ and $\frac{3}{15}$.

Adding the Fractions

Now, we can add the fractions. We have $\frac{5}{15} + \frac{3}{15} = \frac{8}{15}$.

Simplifying the Expression

Now, we can simplify the expression by substituting the value of the exponent. We have $z^{\frac{8}{15}}$.

Conclusion

In this article, we simplified the expression $\frac{z^{\frac{1}{3}}}{z^{-\frac{1}{5}}}$. We used the properties of exponents to rewrite the expression with only positive exponents. We assumed that all variables are positive real numbers.

Final Answer

The final answer is $\boxed{z^{\frac{8}{15}}}$.

Related Topics

If you want to learn more about simplifying expressions with exponents, you can check out the following topics:

• Quotient Rule of Exponents: This rule states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a positive real number and $m$ and $n$ are integers.
• Product Rule of Exponents: This rule states that $a^m \cdot a^n = a^{m+n}$, where $a$ is a positive real number and $m$ and $n$ are integers.
• Exponent Rules: This topic covers the rules for simplifying expressions with exponents, including the quotient rule, product rule, and power rule.

Practice Problems

If you want to practice simplifying expressions with exponents, you can try the following problems:

• Problem 1: Simplify the expression $\frac{x^2}{x^3}$.
• Problem 2: Simplify the expression $\frac{y^4}{y^2}$.
• Problem 3: Simplify the expression $\frac{z^6}{z^3}$.

Solutions

Here are the solutions to the practice problems:

• Problem 1: The solution is $x^{-1}$.
• Problem 2: The solution is $y^2$.
• Problem 3: The solution is $z^3$.

Conclusion

In this article, we simplified the expression $\frac{z^{\frac{1}{3}}}{z^{-\frac{1}{5}}}$. We used the properties of exponents to rewrite the expression with only positive exponents. We assumed that all variables are positive real numbers. We also provided practice problems and solutions to help you practice simplifying expressions with exponents.

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Introduction

In this article, we will provide a Q&A section to help you understand the concept of simplifying expressions with exponents. We will cover common questions and answers related to the topic.

Q&A

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a positive real number and $m$ and $n$ are integers.

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

A: To simplify an expression with a negative exponent, you can use the quotient rule of exponents. For example, $\frac{z^{\frac{1}{3}}}{z^{-\frac{1}{5}}}$ can be simplified as $z^{\frac{1}{3} - (-\frac{1}{5})}$.

Q: What is the product rule of exponents?

A: The product rule of exponents states that $a^m \cdot a^n = a^{m+n}$, where $a$ is a positive real number and $m$ and $n$ are integers.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the product rule of exponents. For example, $z^2 \cdot z^3$ can be simplified as $z^{2+3} = z^5$.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $(a^m)^n = a^{m \cdot n}$, where $a$ is a positive real number and $m$ and $n$ are integers.

Q: How do I simplify an expression with a power of a power?

A: To simplify an expression with a power of a power, you can use the power rule of exponents. For example, $(z^2)^3$ can be simplified as $z^{2 \cdot 3} = z^6$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that $a^0 = 1$, where $a$ is a positive real number.

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

A: To simplify an expression with a zero exponent, you can use the zero exponent rule. For example, $z^0$ can be simplified as $1$.

Conclusion

In this article, we provided a Q&A section to help you understand the concept of simplifying expressions with exponents. We covered common questions and answers related to the topic, including the quotient rule, product rule, power rule, and zero exponent rule.

Final Answer

The final answer is $\boxed{z^{\frac{8}{15}}}$.

Related Topics

If you want to learn more about simplifying expressions with exponents, you can check out the following topics:

• Quotient Rule of Exponents: This rule states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a positive real number and $m$ and $n$ are integers.
• Product Rule of Exponents: This rule states that $a^m \cdot a^n = a^{m+n}$, where $a$ is a positive real number and $m$ and $n$ are integers.
• Power Rule of Exponents: This rule states that $(a^m)^n = a^{m \cdot n}$, where $a$ is a positive real number and $m$ and $n$ are integers.
• Zero Exponent Rule: This rule states that $a^0 = 1$, where $a$ is a positive real number.

Practice Problems

If you want to practice simplifying expressions with exponents, you can try the following problems:

• Problem 1: Simplify the expression $\frac{x^2}{x^3}$.
• Problem 2: Simplify the expression $\frac{y^4}{y^2}$.
• Problem 3: Simplify the expression $\frac{z^6}{z^3}$.

Solutions

Here are the solutions to the practice problems:

• Problem 1: The solution is $x^{-1}$.
• Problem 2: The solution is $y^2$.
• Problem 3: The solution is $z^3$.

Conclusion

In this article, we provided a Q&A section to help you understand the concept of simplifying expressions with exponents. We covered common questions and answers related to the topic, including the quotient rule, product rule, power rule, and zero exponent rule. We also provided practice problems and solutions to help you practice simplifying expressions with exponents.