Simplify And Write The Result Without Negative Exponents. Assume No Variable Is 0. A − 6 A − 7 A − 15 \frac{a^{-6} A^{-7}}{a^{-15}} A − 15 A − 6 A − 7 ​

Understanding Exponents and Negative Exponents

In mathematics, exponents are a shorthand way of representing repeated multiplication of a number. For example, $a^3$ can be read as "a to the power of 3" and is equivalent to $a \times a \times a$. Negative exponents, on the other hand, represent the reciprocal of a number raised to a positive exponent. For instance, $a^{-3}$ is equivalent to $\frac{1}{a^3}$.

Simplifying the Given Expression

The given expression is $\frac{a^{-6} a^{-7}}{a^{-15}}$. Our goal is to simplify this expression and write the result without negative exponents. To do this, we will use the properties of exponents, specifically the rule for multiplying and dividing exponents with the same base.

Applying the Properties of Exponents

When multiplying exponents with the same base, we add the exponents. Conversely, when dividing exponents with the same base, we subtract the exponents. Using these properties, we can simplify the given expression as follows:

$$\frac{a^{-6} a^{-7}}{a^{-15}} = a^{-6-7} \div a^{-15}$$

Simplifying the Expression Further

Now, we can simplify the expression further by applying the rule for dividing exponents with the same base:

$$a^{-6-7} \div a^{-15} = a^{-13} \div a^{-15}$$

Using the Quotient Rule for Exponents

The quotient rule for exponents states that when dividing exponents with the same base, we subtract the exponents. Applying this rule, we get:

$$a^{-13} \div a^{-15} = a^{-13+15}$$

Simplifying the Final Expression

Now, we can simplify the final expression by combining the exponents:

$$a^{-13+15} = a^2$$

Conclusion

In this article, we simplified the given expression $\frac{a^{-6} a^{-7}}{a^{-15}}$ using the properties of exponents. We applied the rules for multiplying and dividing exponents with the same base to arrive at the final simplified expression, $a^2$. This result demonstrates the importance of understanding and applying the properties of exponents in algebraic manipulations.

Common Mistakes to Avoid

When simplifying expressions with negative exponents, it's essential to remember the following:

• Negative exponents represent the reciprocal of a number raised to a positive exponent.
• When multiplying exponents with the same base, add the exponents.
• When dividing exponents with the same base, subtract the exponents.
• Be careful when applying the quotient rule for exponents, as it involves subtracting the exponents.

Real-World Applications

Understanding and applying the properties of exponents has numerous real-world applications in various fields, including:

• Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and physical phenomena.
• Engineering: Exponents are used to model complex systems, such as electrical circuits and mechanical systems.
• Finance: Exponents are used to calculate compound interest and investment returns.

Final Thoughts

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, we add the exponents. For example, $a^3 \times a^4 = a^{3+4} = a^7$.

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

A: When dividing exponents with the same base, we subtract the exponents. For example, $a^3 \div a^4 = a^{3-4} = a^{-1}$.

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

A: To simplify an expression with a negative exponent, we can rewrite it as a fraction with a positive exponent. For example, $a^{-3} = \frac{1}{a^3}$.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing exponents with the same base, we subtract the exponents. For example, $a^m \div a^n = a^{m-n}$.

Q: Can I simplify an expression with multiple negative exponents?

A: Yes, you can simplify an expression with multiple negative exponents by applying the rules for multiplying and dividing exponents with the same base. For example, $\frac{a^{-3} a^{-4}}{a^{-5}} = a^{-3-4} \div a^{-5} = a^{-7} \div a^{-5} = a^{-7+5} = a^{-2}$.

Q: How do I apply the properties of exponents to simplify an expression?

A: To apply the properties of exponents to simplify an expression, follow these steps:

1. Identify the base and exponents in the expression.
2. Apply the rules for multiplying and dividing exponents with the same base.
3. Simplify the expression by combining like terms.
4. Check your work by plugging in values for the base and exponents.

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Forgetting to apply the rules for multiplying and dividing exponents with the same base.
• Not simplifying the expression by combining like terms.
• Not checking your work by plugging in values for the base and exponents.

Q: How do I use exponents in real-world applications?

A: Exponents are used in various real-world applications, including:

• Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and physical phenomena.
• Engineering: Exponents are used to model complex systems, such as electrical circuits and mechanical systems.
• Finance: Exponents are used to calculate compound interest and investment returns.

Q: What are some examples of expressions that can be simplified using exponents?

A: Some examples of expressions that can be simplified using exponents include:

• $\frac{a^3 a^4}{a^5} = a^{3+4-5} = a^2$
• $\frac{a^{-3} a^{-4}}{a^{-5}} = a^{-3-4+5} = a^{-2}$
• $a^3 \times a^4 \times a^5 = a^{3+4+5} = a^{12}$

Q: How do I practice simplifying expressions with exponents?

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

• Start with simple expressions and gradually move on to more complex ones.
• Use online resources, such as calculators and worksheets, to practice simplifying expressions with exponents.
• Work with a partner or tutor to get feedback on your work and improve your skills.
• Take online courses or watch video tutorials to learn more about exponents and how to simplify expressions with them.