Simplify The Expression: ( − 2 Q 5 ) ( − 6 A 3 ) 3 6 A 4 \frac{\left(-2 Q^5\right)\left(-6 A^3\right)^3}{6 A^4} 6 A 4 ( − 2 Q 5 ) ( − 6 A 3 ) 3

Mar 12, 2025 by ADMIN 146 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

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Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying the given expression: $\frac{\left(-2 q^5\right)\left(-6 a^3\right)^3}{6 a^4}$ . We will break down the expression into manageable parts, apply the rules of exponents, and simplify the resulting expression.

Understanding the Expression

The given expression is a fraction, with the numerator being the product of two terms: $\left(-2 q^5\right)$ and $\left(-6 a^3\right)^3$ . The denominator is a single term: $6 a^4$ . To simplify the expression, we need to apply the rules of exponents and manipulate the terms to reveal their underlying structure.

Applying the Rules of Exponents

The first step in simplifying the expression is to apply the rules of exponents. The expression contains two terms with exponents: $\left(-2 q^5\right)$ and $\left(-6 a^3\right)^3$ . We can start by simplifying the second term using the power rule of exponents, which states that $\left(a^m\right)^n = a^{m \cdot n}$ .

import sympy as sp
q = sp.symbols('q')
a = sp.symbols('a')

expr = (-2 * q5) * (-6 * a3)3 / (6 * a4)

simplified_expr = sp.simplify(expr)
print(simplified_expr)

Simplifying the Numerator

The numerator of the expression is the product of two terms: $\left(-2 q^5\right)$ and $\left(-6 a^3\right)^3$ . We can simplify the numerator by applying the product rule of exponents, which states that $a^m \cdot a^n = a^{m + n}$ .

# Simplify the numerator using the product rule of exponents
numerator = (-2 * q**5) * (-6 * a**3)**3
simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

Simplifying the Denominator

The denominator of the expression is a single term: $6 a^4$ . We can simplify the denominator by factoring out the greatest common factor (GCF) of the coefficients and the variables.

# Simplify the denominator by factoring out the GCF
denominator = 6 * a**4
simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

Combining the Simplified Terms

Now that we have simplified the numerator and the denominator, we can combine the simplified terms to reveal the underlying structure of the expression.

# Combine the simplified terms
simplified_expr = simplified_numerator / simplified_denominator
print(simplified_expr)

Conclusion

In this article, we simplified the given expression: $\frac{\left(-2 q^5\right)\left(-6 a^3\right)^3}{6 a^4}$ . We applied the rules of exponents, manipulated the terms, and simplified the resulting expression. The final simplified expression is:

\frac{-108 q^5 a^{12}}{6 a^4}

This expression reveals the underlying structure of the original expression, making it easier to understand and work with.

Final Answer

The final answer is: $\boxed{\frac{-108 q^5 a^{12}}{6 a^4}}$

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Introduction

In our previous article, we simplified the given expression: $\frac{\left(-2 q^5\right)\left(-6 a^3\right)^3}{6 a^4}$ . We applied the rules of exponents, manipulated the terms, and simplified the resulting expression. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern the behavior of exponents in algebraic expressions. The main rules of exponents are:

Product Rule: $a^m \cdot a^n = a^{m + n}$
Power Rule: $\left(a^m\right)^n = a^{m \cdot n}$
Quotient Rule: $\frac{a^m}{a^n} = a^{m - n}$

Q: How do I apply the rules of exponents?

A: To apply the rules of exponents, you need to identify the exponents in the expression and then apply the corresponding rule. For example, if you have the expression $a^3 \cdot a^4$ , you can apply the product rule to simplify it to $a^{3 + 4} = a^7$ .

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same. For example, in the expression $2x + 3$ , $x$ is a variable and $3$ is a constant.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and the denominator and divide both by the GCF. For example, if you have the fraction $\frac{12}{18}$ , you can simplify it by dividing both the numerator and the denominator by $6$ to get $\frac{2}{3}$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that govern the order in which you perform operations in an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to simplify the numerator and the denominator separately and then simplify the resulting fraction. For example, if you have the complex fraction $\frac{\frac{1}{2}}{\frac{3}{4}}$ , you can simplify it by simplifying the numerator and the denominator separately and then simplifying the resulting fraction.

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying algebraic expressions. We covered the rules of exponents, how to apply them, the difference between variables and constants, how to simplify fractions, the order of operations, and how to simplify complex fractions. By following these rules and techniques, you can simplify even the most complex algebraic expressions.

Final Answer

The final answer is: There is no final numerical answer to this article. The article is a Q&A guide to algebraic manipulation.