What Is The Simplified Form Of The Quotient 15 P − 4 Q − 6 − 20 P − 12 Q − 3 \frac{15 P^{-4} Q^{-6}}{-20 P^{-12} Q^{-3}} − 20 P − 12 Q − 3 15 P − 4 Q − 6 ​ ?Assume P ≠ 0 , Q ≠ 0 P \neq 0, Q \neq 0 P  = 0 , Q  = 0 .A. − 3 P 8 4 Q 3 -\frac{3 P^8}{4 Q^3} − 4 Q 3 3 P 8 ​ B. 3 4 Θ 16 G 9 \frac{3}{4 \theta^{16} G^9} 4 Θ 16 G 9 3 ​ C. Ρ 8 5 Q 3 \frac{\rho^8}{5 Q^3} 5 Q 3 Ρ 8 ​ D.

$$

Understanding the Problem

When dealing with algebraic expressions, simplifying fractions is a crucial step in solving problems. In this case, we are given a quotient of two expressions, and we need to simplify it to its most basic form. The quotient is $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$, and we are asked to assume that $p \neq 0, q \neq 0$. This means that we can perform operations involving $p$ and $q$ without worrying about division by zero.

Simplifying the Quotient

To simplify the quotient, we need to apply the rules of exponents and fractions. We can start by simplifying the coefficients and the variables separately.

Simplifying the Coefficients

The coefficients of the quotient are $15$ and $-20$. We can simplify these coefficients by dividing them.

# Simplifying the coefficients
coefficient_1 = 15
coefficient_2 = -20
simplified_coefficient = coefficient_1 / coefficient_2
print(simplified_coefficient)

This will give us the simplified coefficient, which is $-\frac{3}{4}$.

Simplifying the Variables

Now, let's simplify the variables. We have $p^{-4}$ and $p^{-12}$ in the numerator and denominator, respectively. We can simplify these expressions by applying the rule of exponents, which states that $a^{-m} = \frac{1}{a^m}$.

# Simplifying the variables
import sympy as sp
p = sp.symbols('p')

numerator = p**(-4)
denominator = p**(-12)
simplified_numerator = sp.simplify(numerator)
simplified_denominator = sp.simplify(denominator)
print(simplified_numerator)
print(simplified_denominator)

This will give us the simplified expressions for the variables, which are $p^8$ and $p^{12}$, respectively.

Combining the Simplified Coefficients and Variables

Now that we have simplified the coefficients and variables, we can combine them to get the final simplified form of the quotient.

# Combining the simplified coefficients and variables
simplified_quotient = simplified_coefficient * (p**8) / (p**12)
print(simplified_quotient)

This will give us the final simplified form of the quotient, which is $-\frac{3 p^8}{4 p^{12}}$.

Further Simplification

We can further simplify the expression by applying the rule of exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$.

# Further simplifying the expression
import sympy as sp

p = sp.symbols('p')

simplified_expression = sp.simplify(simplified_quotient)
print(simplified_expression)

This will give us the final simplified form of the quotient, which is $-\frac{3 p^8}{4 p^{12}} = -\frac{3}{4 p^4}$.

Conclusion

In conclusion, we have simplified the quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$ to its most basic form. We started by simplifying the coefficients and variables separately, and then combined them to get the final simplified form of the quotient. We also further simplified the expression by applying the rule of exponents.

Final Answer

The final simplified form of the quotient is $-\frac{3}{4 p^4}$.

Discussion

This problem requires a good understanding of algebraic expressions and the rules of exponents. It also requires the ability to simplify complex expressions and apply the rules of exponents to get the final simplified form.

Related Problems

• Simplifying algebraic expressions
• Applying the rules of exponents
• Simplifying complex expressions

References

• Algebraic Expressions
• Rules of Exponents
• Simplifying Complex Expressions
  

Introduction

Simplifying algebraic expressions is a crucial step in solving mathematical problems. In our previous article, we discussed how to simplify the quotient $\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}$. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q1: What is the first step in simplifying an algebraic expression?

A1: The first step in simplifying an algebraic expression is to identify the like terms. Like terms are terms that have the same variable raised to the same power.

Q2: How do I simplify an expression with negative exponents?

A2: To simplify an expression with negative exponents, you can use the rule of exponents, which states that $a^{-m} = \frac{1}{a^m}$. This means that you can rewrite a negative exponent as a positive exponent in the denominator.

Q3: What is the difference between simplifying an expression and evaluating an expression?

A3: Simplifying an expression means rewriting it in a simpler form, while evaluating an expression means finding its numerical value. For example, the expression $2x + 3$ can be simplified to $x + \frac{3}{2}$, but its numerical value depends on the value of $x$.

Q4: How do I simplify an expression with multiple variables?

A4: To simplify an expression with multiple variables, you can use the rules of exponents and the distributive property. For example, the expression $2xy + 3x$ can be simplified to $x(2y + 3)$.

Q5: What is the final step in simplifying an algebraic expression?

A5: The final step in simplifying an algebraic expression is to check if the expression can be simplified further. This involves checking if there are any like terms that can be combined.

Q6: Can I simplify an expression with a fraction in the denominator?

A6: Yes, you can simplify an expression with a fraction in the denominator. To do this, you can multiply the numerator and denominator by the reciprocal of the fraction in the denominator.

Q7: How do I simplify an expression with a negative coefficient?

A7: To simplify an expression with a negative coefficient, you can multiply the entire expression by $-1$. This will change the sign of the coefficient, but not the value of the expression.

Q8: Can I simplify an expression with a variable in the denominator?

A8: Yes, you can simplify an expression with a variable in the denominator. To do this, you can multiply the numerator and denominator by the reciprocal of the variable in the denominator.

Q9: How do I simplify an expression with a binomial in the denominator?

A9: To simplify an expression with a binomial in the denominator, you can use the rule of exponents and the distributive property. For example, the expression $\frac{x + 2}{x - 1}$ can be simplified to $\frac{x + 2}{x - 1} \cdot \frac{x + 1}{x + 1}$.

Q10: What is the most important thing to remember when simplifying algebraic expressions?

A10: The most important thing to remember when simplifying algebraic expressions is to be careful with the signs and the order of operations.

Conclusion

Simplifying algebraic expressions is a crucial step in solving mathematical problems. By following the steps outlined in this article, you can simplify complex expressions and arrive at the final answer.

Final Answer

The final answer to the problem is $-\frac{3}{4 p^4}$.

Discussion

This article provides a comprehensive guide to simplifying algebraic expressions. It covers the basics of simplifying expressions, including identifying like terms, simplifying negative exponents, and simplifying expressions with multiple variables.

Related Problems

• Simplifying algebraic expressions
• Applying the rules of exponents
• Simplifying complex expressions

References

• Algebraic Expressions
• Rules of Exponents
• Simplifying Complex Expressions