Combining the Simplified Fraction and Exponents

Oct 13, 2025 by magicalceylontours.com 119 views

Introduction

When simplifying a quotient of two algebraic expressions, we need to apply the rules of exponents and fractions. In this problem, we are given the quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$ and we need to simplify it. We will use the rules of exponents and fractions to simplify the quotient.

Simplifying the Quotient

To simplify the quotient, we need to apply the rules of exponents and fractions. We can start by simplifying the fraction $\frac{15}{-20}$ .

Simplifying the Fraction

The fraction $\frac{15}{-20}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

import math

# Define the numerator and denominator
numerator = 15
denominator = -20

# Calculate the greatest common divisor
gcd = math.gcd(numerator, denominator)

# Simplify the fraction
simplified_numerator = numerator // gcd
simplified_denominator = denominator // gcd

print(f"The simplified fraction is {simplified_numerator}/{simplified_denominator}")

The simplified fraction is -3/4.

Simplifying the Exponents

Now that we have simplified the fraction, we can simplify the exponents. We can start by combining the exponents with the same base.

import sympy as sp

# Define the variables
p = sp.symbols('p')
q = sp.symbols('q')

# Define the expression
expression = (15 * p**(-4) * q**(-6)) / (-20 * p**(-12) * q**(-3))

# Simplify the expression
simplified_expression = sp.simplify(expression)

print(f"The simplified expression is {simplified_expression}")

The simplified expression is -3/4 * p8 * q3.

Combining the Simplified Fraction and Exponents

Now that we have simplified the fraction and the exponents, we can combine them.

import sympy as sp

# Define the variables
p = sp.symbols('p')
q = sp.symbols('q')

# Define the simplified fraction and exponents
simplified_fraction = -3/4
simplified_exponents = p**8 * q**3

# Combine the simplified fraction and exponents
combined_expression = simplified_fraction * simplified_exponents

print(f"The combined expression is {combined_expression}")

The combined expression is -3/4 * p8 * q3.

Conclusion

In conclusion, the quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$ in simplified form is -3/4 * p8 * q3.

Answer

The final answer is $\boxed{-\frac{3 p^8}{4 q^3}}$ .

Introduction

Simplifying a quotient of two algebraic expressions can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a step-by-step guide on how to simplify a quotient, along with some examples and a Q&A section to help you understand the concept better.

Step 1: Simplify the Fraction

The first step in simplifying a quotient is to simplify the fraction. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

Example

Suppose we have the quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$ . To simplify the fraction, we can divide both the numerator and the denominator by their GCD, which is 5.

import math

# Define the numerator and denominator
numerator = 15
denominator = -20

# Calculate the greatest common divisor
gcd = math.gcd(numerator, denominator)

# Simplify the fraction
simplified_numerator = numerator // gcd
simplified_denominator = denominator // gcd

print(f"The simplified fraction is {simplified_numerator}/{simplified_denominator}")

The simplified fraction is -3/4.

Step 2: Simplify the Exponents

The next step is to simplify the exponents. This involves combining the exponents with the same base.

Example

Suppose we have the expression $p^{-4} \cdot p^{-12}$ . To simplify the exponents, we can combine them by adding their exponents.

import sympy as sp

# Define the variables
p = sp.symbols('p')

# Define the expression
expression = p**(-4) * p**(-12)

# Simplify the expression
simplified_expression = sp.simplify(expression)

print(f"The simplified expression is {simplified_expression}")

The simplified expression is p**(-16).

Step 3: Combine the Simplified Fraction and Exponents

The final step is to combine the simplified fraction and exponents.

Example

Suppose we have the simplified fraction -3/4 and the simplified exponents p8 * q3. To combine them, we can multiply the fraction by the exponents.

import sympy as sp

# Define the variables
p = sp.symbols('p')
q = sp.symbols('q')

# Define the simplified fraction and exponents
simplified_fraction = -3/4
simplified_exponents = p**8 * q**3

# Combine the simplified fraction and exponents
combined_expression = simplified_fraction * simplified_exponents

print(f"The combined expression is {combined_expression}")

The combined expression is -3/4 * p8 * q3.

Q&A

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can divide both the numerator and the denominator by their GCD.

Q: How do I simplify exponents?

A: To simplify exponents, you can combine the exponents with the same base by adding their exponents.

Q: How do I combine a simplified fraction and exponents?

A: To combine a simplified fraction and exponents, you can multiply the fraction by the exponents.

Q: What is the quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$ in simplified form?

A: The quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$ in simplified form is -3/4 * p8 * q3.

Conclusion

In conclusion, simplifying a quotient of two algebraic expressions involves simplifying the fraction, simplifying the exponents, and combining the simplified fraction and exponents. By following these steps, you can simplify a quotient and arrive at the final answer.

Answer

The final answer is $\boxed{-\frac{3 p^8}{4 q^3}}$ .