Rule of Dividing Like Bases with Exponents

Oct 13, 2025 by miodeya.com 100 views

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with exponents, we can use the rules of exponentiation to simplify expressions. In this article, we will simplify the expression $\frac{x^{11} y^{10}}{x^{-3} y^{-1}}$ using the rules of exponentiation.

Understanding Exponents

Before we dive into simplifying the expression, let's understand what exponents are. An exponent is a small number that is written above and to the right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $x^3$ , the exponent 3 represents the power to which the base $x$ is raised.

Simplifying the Expression

To simplify the expression $\frac{x^{11} y^{10}}{x^{-3} y^{-1}}$ , we can use the rule of dividing like bases with exponents. When we divide two numbers or variables with the same base and different exponents, we subtract the exponents. In this case, we have:

\frac{x^{11} y^{10}}{x^{-3} y^{-1}} = x^{11-(-3)} y^{10-(-1)}

Applying the Rules of Exponentiation

Now, let's apply the rules of exponentiation to simplify the expression further. When we subtract a negative exponent, we can change the sign of the exponent and add it to the other exponent. In this case, we have:

x^{11-(-3)} y^{10-(-1)} = x^{11+3} y^{10+1}

Simplifying the Expression Further

Now, let's simplify the expression further by combining the exponents. When we add exponents with the same base, we can multiply the coefficients and add the exponents. In this case, we have:

x^{11+3} y^{10+1} = x^{14} y^{11}

Conclusion

In this article, we simplified the expression $\frac{x^{11} y^{10}}{x^{-3} y^{-1}}$ using the rules of exponentiation. We used the rule of dividing like bases with exponents and applied the rules of exponentiation to simplify the expression further. The final simplified expression is $x^{14} y^{11}$ .

Frequently Asked Questions

Q: What is the rule of dividing like bases with exponents?

A: The rule of dividing like bases with exponents states that when we divide two numbers or variables with the same base and different exponents, we subtract the exponents.

Q: How do we simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, we can change the sign of the exponent and add it to the other exponent.

Q: What is the final simplified expression?

A: The final simplified expression is $x^{14} y^{11}$ .

Step-by-Step Solution

Step 1: Understand the expression

The expression is $\frac{x^{11} y^{10}}{x^{-3} y^{-1}}$ .

Step 2: Apply the rule of dividing like bases with exponents

We subtract the exponents: $x^{11-(-3)} y^{10-(-1)}$ .

Step 3: Simplify the expression further

We change the sign of the exponent and add it to the other exponent: $x^{11+3} y^{10+1}$ .

Step 4: Combine the exponents

We multiply the coefficients and add the exponents: $x^{14} y^{11}$ .

Example Problems

Problem 1

Simplify the expression $\frac{x^5 y^3}{x^2 y^1}$ .

Solution

We apply the rule of dividing like bases with exponents: $x^{5-2} y^{3-1} = x^3 y^2$ .

Problem 2

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

Solution

We apply the rule of dividing like bases with exponents: $x^{-2-3} y^{-4-2} = x^{-5} y^{-6}$ .

Practice Problems

Problem 1

Simplify the expression $\frac{x^7 y^9}{x^4 y^5}$ .

Problem 2

Simplify the expression $\frac{x^{-1} y^{-3}}{x^2 y^4}$ .

Problem 3

Simplify the expression $\frac{x^{10} y^{12}}{x^5 y^8}$ .

Problem 4

Simplify the expression $\frac{x^{-6} y^{-9}}{x^3 y^2}$ .

Problem 5

Simplify the expression $\frac{x^{15} y^{18}}{x^9 y^{12}}$ .

Conclusion

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Introduction

In our previous article, we simplified the expression $\frac{x^{11} y^{10}}{x^{-3} y^{-1}}$ 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 like bases with exponents?

A: The rule of dividing like bases with exponents states that when we divide two numbers or variables with the same base and different exponents, we subtract the exponents.

Q: How do we simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, we can change the sign of the exponent and add it to the other exponent.

Q: What is the final simplified expression?

A: The final simplified expression is $x^{14} y^{11}$ .

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

A: Yes, we can simplify expressions with exponents that have different bases by using the rule of dividing like bases with exponents.

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

A: An expression with a zero exponent is equal to 1. For example, $x^0 = 1$ .

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

A: Yes, we can simplify expressions with exponents that have fractional exponents by using the rule of dividing like bases with exponents.

Q: How do we simplify an expression with a negative exponent and a fractional exponent?

A: To simplify an expression with a negative exponent and a fractional exponent, we can change the sign of the exponent and add it to the other exponent.

Q: Can we simplify expressions with exponents that have imaginary exponents?

A: Yes, we can simplify expressions with exponents that have imaginary exponents by using the rule of dividing like bases with exponents.

Q: How do we simplify an expression with a complex exponent?

A: To simplify an expression with a complex exponent, we can use the rule of dividing like bases with exponents and the properties of complex numbers.

Example Problems

Problem 1

Simplify the expression $\frac{x^5 y^3}{x^2 y^1}$ .

Solution

We apply the rule of dividing like bases with exponents: $x^{5-2} y^{3-1} = x^3 y^2$ .

Problem 2

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

Solution

We apply the rule of dividing like bases with exponents: $x^{-2-3} y^{-4-2} = x^{-5} y^{-6}$ .

Problem 3

Simplify the expression $\frac{x^{10} y^{12}}{x^5 y^8}$ .

Solution

We apply the rule of dividing like bases with exponents: $x^{10-5} y^{12-8} = x^5 y^4$ .

Practice Problems

Problem 1

Simplify the expression $\frac{x^7 y^9}{x^4 y^5}$ .

Problem 2

Simplify the expression $\frac{x^{-1} y^{-3}}{x^2 y^4}$ .

Problem 3

Simplify the expression $\frac{x^{15} y^{18}}{x^9 y^{12}}$ .

Problem 4

Simplify the expression $\frac{x^{-6} y^{-9}}{x^3 y^2}$ .

Problem 5

Simplify the expression $\frac{x^{20} y^{24}}{x^10 y^{16}}$ .

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with exponents. We also provided example problems and practice problems to help you practice simplifying expressions with exponents. Remember to use the rule of dividing like bases with exponents and the properties of exponents to simplify expressions with exponents.

Additional Resources

Glossary

Exponent: A small number that is written above and to the right of a number or a variable. It represents the power to which the base is raised.
Base: The number or variable that is being raised to a power.
Power: The result of raising a base to an exponent.
Rule of Dividing Like Bases with Exponents: When we divide two numbers or variables with the same base and different exponents, we subtract the exponents.