Simplify $ \frac{\left(x^{-3} X^5\right)(3 Y) 3}{x {-1} X^2} $.A. $ 27 X^2 Y^3 $B. $ 9 X Y^3 $C. $ 27 X Y^3 $D. $ 9 X^3 Y $

Mar 11, 2025 by ADMIN 124 views

Simplify $ \frac{\left(x^{-3} x^5\right)(3 y)^3}{x{-1} x^2} $

Understanding Exponents and Simplifying Expressions

When dealing with exponents, it's essential to understand the rules of exponentiation. The product rule states that when multiplying two powers with the same base, we add the exponents. The quotient rule states that when dividing two powers with the same base, we subtract the exponents. Additionally, the power rule states that when raising a power to another power, we multiply the exponents.

In the given expression, we have several terms with exponents that need to be simplified. Let's start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator is given by $\left(x^{-3} x^5\right)(3 y)^3$ . To simplify this expression, we can use the product rule for exponents. When multiplying two powers with the same base, we add the exponents.

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

numerator = (x**-3 * x**5) * (3*y)**3

simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

The simplified numerator is $x^2(3y)^3$ . Now, let's simplify the denominator.

Simplifying the Denominator

The denominator is given by $x^{-1} x^2$ . To simplify this expression, we can use the quotient rule for exponents. When dividing two powers with the same base, we subtract the exponents.

# Define the denominator
denominator = x**-1 * x**2

simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

The simplified denominator is $x^1$ or simply $x$ .

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can simplify the entire expression by dividing the numerator by the denominator.

# Simplify the expression
simplified_expression = sp.simplify(simplified_numerator / simplified_denominator)
print(simplified_expression)

The simplified expression is $x^2(3y)^3/x = 27x^2y^3$ .

Conclusion

In conclusion, the simplified expression is $27x^2y^3$ . This is the correct answer.

Answer

The correct answer is A. $27 x^2 y^3$ .

Discussion

This problem requires a good understanding of exponents and simplifying expressions. The product rule, quotient rule, and power rule for exponents are essential concepts to master when dealing with expressions involving exponents.

Tips and Tricks

When simplifying expressions involving exponents, always start by simplifying the numerator and denominator separately.
Use the product rule, quotient rule, and power rule for exponents to simplify expressions.
Make sure to simplify the expression by dividing the numerator by the denominator.

Practice Problems

Simplify the expression $\frac{(x^2y^3)^2}{x^3y^2}$ .
Simplify the expression $\frac{x^5y^2}{x^2y^3}$ .
Simplify the expression $\frac{(x^2y^3)^3}{x^4y^2}$ .

Solutions

The simplified expression is $x^4y^6/x^3y^2 = x^1y^4 = xy^4$ .
The simplified expression is $x^3y^2/x^2y^3 = x^1y^2/x^2y^3 = 1/y$ .
The simplified expression is $(x^2y^3)^3/x^4y^2 = x^6y^9/x^4y^2 = x^2y^7$ .

Conclusion

In conclusion, simplifying expressions involving exponents requires a good understanding of the product rule, quotient rule, and power rule for exponents. By mastering these concepts, you can simplify complex expressions and arrive at the correct answer.
Simplify $ \frac{\left(x^{-3} x^5\right)(3 y)^3}{x{-1} x^2} $: Q&A

Q: What is the product rule for exponents?

A: The product rule for exponents states that when multiplying two powers with the same base, we add the exponents. For example, $x^a \cdot x^b = x^{a+b}$ .

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents. For example, $x^a \div x^b = x^{a-b}$ .

Q: What is the power rule for exponents?

A: The power rule for exponents states that when raising a power to another power, we multiply the exponents. For example, $(x^a)^b = x^{ab}$ .

Q: How do I simplify the expression $\frac{(x^2y^3)^2}{x^3y^2}$ ?

A: To simplify this expression, we can use the product rule, quotient rule, and power rule for exponents. First, we simplify the numerator using the product rule: $(x^2y^3)^2 = x^4y^6$ . Then, we simplify the denominator using the quotient rule: $x^3y^2 \div x^3y^2 = 1$ . Finally, we simplify the expression by dividing the numerator by the denominator: $x^4y^6 \div 1 = x^4y^6$ .

Q: How do I simplify the expression $\frac{x^5y^2}{x^2y^3}$ ?

A: To simplify this expression, we can use the quotient rule for exponents. We subtract the exponents of the denominator from the exponents of the numerator: $x^5y^2 \div x^2y^3 = x^{5-2}y^{2-3} = x^3y^{-1} = \frac{x^3}{y}$ .

Q: How do I simplify the expression $\frac{(x^2y^3)^3}{x^4y^2}$ ?

A: To simplify this expression, we can use the product rule, quotient rule, and power rule for exponents. First, we simplify the numerator using the product rule: $(x^2y^3)^3 = x^6y^9$ . Then, we simplify the denominator using the quotient rule: $x^4y^2 \div x^4y^2 = 1$ . Finally, we simplify the expression by dividing the numerator by the denominator: $x^6y^9 \div 1 = x^6y^9$ .

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

A: A variable is a symbol that represents a value that can change, such as x or y. A constant is a value that does not change, such as 2 or 5.

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

A: To simplify an expression with multiple variables, we can use the product rule, quotient rule, and power rule for exponents. We simplify each variable separately and then combine the simplified expressions.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate expressions with exponents next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Evaluate addition and subtraction operations from left to right.

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

A: To simplify an expression with negative exponents, we can use the quotient rule for exponents. We subtract the exponents of the denominator from the exponents of the numerator.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is an expression that contains a fraction, such as $\frac{x^2}{y^3}$ . A polynomial expression is an expression that contains only variables and constants, such as $x^2 + 3y^2$ .

Q: How do I simplify a rational expression?

A: To simplify a rational expression, we can use the product rule, quotient rule, and power rule for exponents. We simplify the numerator and denominator separately and then simplify the resulting expression.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression that contains only variables and constants, such as $x^2 + 3y^2$ . A quadratic expression is an expression that contains a squared variable, such as $x^2 + 2xy + y^2$ .

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, we can use the product rule, quotient rule, and power rule for exponents. We simplify the expression by combining like terms and then simplifying the resulting expression.

Conclusion