Select The Correct Answer.Simplify The Following Expression: $6^{\frac{1}{5}} \div 6^{\frac{10}{3}}$A. $\frac{1}{18}$ B. $\frac{1}{216}$ C. 216 D. 18

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the expression $6^{\frac{1}{5}} \div 6^{\frac{10}{3}}$. We will break down the problem step by step, using the properties of exponents to arrive at the correct answer.

Understanding Exponents

Before we dive into the problem, let's review the basics of exponents. An exponent is a small number that is raised to the power of a larger number. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2$. Exponents can also be negative, which means we are dividing by the base number. For example, $2^{-3}$ means $\frac{1}{2^3}$.

The Quotient of Powers Property

One of the most important properties of exponents is the quotient of powers property, which states that when we divide two powers with the same base, we can subtract the exponents. In other words, $\frac{a^m}{a^n} = a^{m-n}$. This property will be crucial in simplifying the given expression.

Simplifying the Expression

Now that we have reviewed the basics of exponents and the quotient of powers property, let's simplify the expression $6^{\frac{1}{5}} \div 6^{\frac{10}{3}}$. Using the quotient of powers property, we can rewrite the expression as:

$$6^{\frac{1}{5}} \div 6^{\frac{10}{3}} = 6^{\frac{1}{5} - \frac{10}{3}}$$

Simplifying the Exponents

Now that we have simplified the expression, let's focus on simplifying the exponents. To do this, we need to find a common denominator for the fractions. The least common multiple of $5$ and $3$ is $15$, so we can rewrite the exponents as:

$$\frac{1}{5} - \frac{10}{3} = \frac{3}{15} - \frac{50}{15} = -\frac{47}{15}$$

The Final Answer

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

$$6^{-\frac{47}{15}}$$

To simplify this expression further, we can use the fact that $6^{-n} = \frac{1}{6^n}$. Therefore, we can rewrite the expression as:

$$6^{-\frac{47}{15}} = \frac{1}{6^{\frac{47}{15}}}$$

Evaluating the Expression

Now that we have simplified the expression, let's evaluate it. To do this, we need to find the value of $6^{\frac{47}{15}}$. We can do this by using a calculator or by simplifying the expression further.

Using a calculator, we can find that $6^{\frac{47}{15}} \approx 18.18$. Therefore, we can rewrite the expression as:

$$\frac{1}{6^{\frac{47}{15}}} \approx \frac{1}{18.18}$$

Conclusion

In this article, we simplified the expression $6^{\frac{1}{5}} \div 6^{\frac{10}{3}}$ using the quotient of powers property and the fact that $6^{-n} = \frac{1}{6^n}$. We found that the final answer is $\frac{1}{18.18}$, which is equivalent to $\frac{1}{18}$.

Discussion

The expression $6^{\frac{1}{5}} \div 6^{\frac{10}{3}}$ is a classic example of a quotient of powers problem. The quotient of powers property is a fundamental concept in mathematics, and it is used extensively in algebra and calculus. In this article, we used the quotient of powers property to simplify the expression and arrive at the correct answer.

Common Mistakes

When simplifying exponential expressions, it is easy to make mistakes. One common mistake is to forget to use the quotient of powers property when dividing two powers with the same base. Another common mistake is to simplify the exponents incorrectly. To avoid these mistakes, it is essential to review the basics of exponents and the quotient of powers property.

Conclusion

$$$$$$$$

Introduction

In our previous article, we simplified the expression $6^{\frac{1}{5}} \div 6^{\frac{10}{3}}$ using the quotient of powers property and the fact that $6^{-n} = \frac{1}{6^n}$. In this article, we will answer some common questions related to simplifying exponential expressions.

Q&A

Q: What is the quotient of powers property?

A: The quotient of powers property states that when we divide two powers with the same base, we can subtract the exponents. In other words, $\frac{a^m}{a^n} = a^{m-n}$.

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 fact that $a^{-n} = \frac{1}{a^n}$. For example, $6^{-3} = \frac{1}{6^3}$.

Q: What is the difference between a quotient of powers and a product of powers?

A: A quotient of powers is when we divide two powers with the same base, while a product of powers is when we multiply two powers with the same base. For example, $\frac{2^3}{2^2}$ is a quotient of powers, while $2^3 \times 2^2$ is a product of powers.

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 fact that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, $6^{\frac{1}{3}} = \sqrt[3]{6}$.

Q: What is the order of operations for simplifying exponential expressions?

A: The order of operations for simplifying exponential expressions is:

1. Evaluate any exponential expressions inside the parentheses.
2. Simplify any negative exponents using the fact that $a^{-n} = \frac{1}{a^n}$.
3. Simplify any fractional exponents using the fact that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.
4. Use the quotient of powers property to simplify any quotients of powers.

Q: How do I check my work when simplifying exponential expressions?

A: To check your work when simplifying exponential expressions, you can use the following steps:

1. Plug in a value for the base and exponent to see if the expression simplifies correctly.
2. Use a calculator to evaluate the expression and see if it matches the simplified expression.
3. Check your work by plugging in different values for the base and exponent to see if the expression simplifies correctly.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill for students and professionals alike. By understanding the quotient of powers property and the fact that $6^{-n} = \frac{1}{6^n}$, we can simplify complex exponential expressions with ease. We hope that this Q&A guide has been helpful in answering some common questions related to simplifying exponential expressions.

Common Mistakes

When simplifying exponential expressions, it is easy to make mistakes. Some common mistakes include:

• Forgetting to use the quotient of powers property when dividing two powers with the same base.
• Simplifying the exponents incorrectly.
• Not checking work when simplifying exponential expressions.

To avoid these mistakes, it is essential to review the basics of exponents and the quotient of powers property.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill for students and professionals alike. By understanding the quotient of powers property and the fact that $6^{-n} = \frac{1}{6^n}$, we can simplify complex exponential expressions with ease. We hope that this Q&A guide has been helpful in answering some common questions related to simplifying exponential expressions.