Simplify The Expression: ( 28 X 3 Y 2 2 X Y − 3 ) 2 \left(\frac{28 X^3 Y^2}{2 X Y^{-3}}\right)^2 ( 2 X Y − 3 28 X 3 Y 2 ​ ) 2

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression $\left(\frac{28 x^3 y^2}{2 x y^{-3}}\right)^2$. We will break down the expression into smaller parts, apply the rules of exponents, and simplify the resulting expression.

Understanding the Expression

The given expression is a fraction raised to the power of 2. To simplify this expression, we need to start by simplifying the fraction inside the parentheses. The fraction is $\frac{28 x^3 y^2}{2 x y^{-3}}$. We can simplify this fraction by canceling out common factors in the numerator and denominator.

Simplifying the Fraction

To simplify the fraction, we need to look for common factors in the numerator and denominator. The numerator is $28 x^3 y^2$, and the denominator is $2 x y^{-3}$. We can see that both the numerator and denominator have a common factor of $2 x$. We can cancel out this common factor by dividing both the numerator and denominator by $2 x$.

import sympy as sp
x = sp.symbols('x')
y = sp.symbols('y')

expr = (28x**3y2) / (2xy(-3))

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

The simplified expression is $14 x^2 y^5$.

Applying the Power Rule

Now that we have simplified the fraction, we can apply the power rule to simplify the expression. The power rule states that for any variables $a$ and $b$ and any integer $n$, $(ab)^n = a^nb^n$. We can apply this rule to our expression by raising the simplified fraction to the power of 2.

Simplifying the Expression

To simplify the expression, we need to apply the power rule to the simplified fraction. The simplified fraction is $14 x^2 y^5$, and we need to raise it to the power of 2. We can do this by squaring the coefficient and multiplying the exponents of the variables.

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

simplified_expr = 14x**2y**5

result = simplified_expr**2
print(result)

The resulting expression is $196 x^4 y^{10}$.

Conclusion

In this article, we simplified the expression $\left(\frac{28 x^3 y^2}{2 x y^{-3}}\right)^2$ by breaking it down into smaller parts, applying the rules of exponents, and simplifying the resulting expression. We started by simplifying the fraction inside the parentheses, then applied the power rule to simplify the expression. The final simplified expression is $196 x^4 y^{10}$.

Frequently Asked Questions

• Q: What is the power rule in algebra? A: The power rule states that for any variables $a$ and $b$ and any integer $n$, $(ab)^n = a^nb^n$.
• Q: How do I simplify a fraction raised to a power? A: To simplify a fraction raised to a power, you need to simplify the fraction inside the parentheses, then apply the power rule to simplify the expression.
• Q: What is the resulting expression after simplifying the given expression? A: The resulting expression is $196 x^4 y^{10}$.

Further Reading

• Simplifying Algebraic Expressions: A Guide to the Rules and Techniques
• Understanding Exponents and Powers in Algebra
• Simplifying Fractions and Rational Expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Algebra and Trigonometry" by James Stewart
  

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Introduction

In our previous article, we simplified the expression $\left(\frac{28 x^3 y^2}{2 x y^{-3}}\right)^2$ by breaking it down into smaller parts, applying the rules of exponents, and simplifying the resulting expression. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the power rule in algebra?

A: The power rule states that for any variables $a$ and $b$ and any integer $n$, $(ab)^n = a^nb^n$. This rule allows us to simplify expressions by raising the product of two variables to a power.

Q: How do I simplify a fraction raised to a power?

A: To simplify a fraction raised to a power, you need to simplify the fraction inside the parentheses, then apply the power rule to simplify the expression. This involves canceling out common factors in the numerator and denominator, then raising the resulting fraction to the power.

Q: What is the resulting expression after simplifying the given expression?

A: The resulting expression is $196 x^4 y^{10}$.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents by using the rule $a^{-n} = \frac{1}{a^n}$. This rule allows you to rewrite negative exponents as fractions.

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

A: To simplify an expression with multiple variables, you need to apply the rules of exponents and simplify the expression step by step. This involves canceling out common factors, applying the power rule, and simplifying the resulting expression.

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

A: Yes, you can simplify an expression with a variable in the denominator by using the rule $\frac{1}{a^n} = a^{-n}$. This rule allows you to rewrite fractions with variables in the denominator as expressions with negative exponents.

Q: How do I simplify an expression with a coefficient?

A: To simplify an expression with a coefficient, you need to apply the rules of exponents and simplify the expression step by step. This involves canceling out common factors, applying the power rule, and simplifying the resulting expression.

Examples

Example 1: Simplifying an Expression with Negative Exponents

Simplify the expression $\left(\frac{2 x^3 y^{-2}}{x^2 y^3}\right)^2$.

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

expr = (2x**3y**(-2)) / (x2*y3)

simplified_expr = sp.simplify(expr)

result = simplified_expr**2
print(result)

The resulting expression is $\frac{4 x^4 y^{-4}}{x^4 y^6}$.

Example 2: Simplifying an Expression with Multiple Variables

Simplify the expression $\left(\frac{3 x^2 y^3 z^4}{2 x y^2 z^3}\right)^2$.

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')
z = sp.symbols('z')

expr = (3x**2y3*z4) / (2xy2*z3)

simplified_expr = sp.simplify(expr)

result = simplified_expr**2
print(result)

The resulting expression is $\frac{9 x^4 y^6 z^8}{4 x^4 y^4 z^6}$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We covered topics such as the power rule, simplifying fractions raised to a power, and simplifying expressions with negative exponents. We also provided examples to illustrate the concepts.

Further Reading

• Simplifying Algebraic Expressions: A Guide to the Rules and Techniques
• Understanding Exponents and Powers in Algebra
• Simplifying Fractions and Rational Expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Algebra and Trigonometry" by James Stewart