Apply the product of powers rule

Oct 13, 2025 by talynt.org 91 views

Understanding the Problem

When simplifying algebraic expressions, it's essential to understand the rules of exponents and how to manipulate them to obtain the desired result. In this article, we will focus on simplifying a given expression and selecting the correct equivalent expression from the options provided.

The Given Expression

The given expression is $\frac{14 x^4 y^6}{7 x^6 y^2}$ . Our goal is to simplify this expression and find an equivalent expression from the options provided.

Step 1: Simplify the Numerator and Denominator

To simplify the expression, we can start by simplifying the numerator and denominator separately. The numerator is $14 x^4 y^6$ , and the denominator is $7 x^6 y^2$ .

import sympy as sp
x, y = sp.symbols('x y')
numerator = 14 * x4 * y6
denominator = 7 * x6 * y2
simplified_numerator = sp.simplify(numerator)
simplified_denominator = sp.simplify(denominator)
print(f"Simplified Numerator: simplified_numerator}")
print(f"Simplified Denominator {simplified_denominator")

Step 2: Cancel Out Common Factors

After simplifying the numerator and denominator, we can cancel out any common factors to obtain the simplified expression.

# Cancel out common factors
simplified_expression = sp.cancel(simplified_numerator / simplified_denominator)
print(f"Simplified Expression: {simplified_expression}")

Step 3: Select the Correct Equivalent Expression

Now that we have the simplified expression, we can compare it with the options provided to select the correct equivalent expression.

Option A: $\frac{2 y^4}{x^4}$

Let's compare the simplified expression with Option A.

# Define Option A
option_a = 2 * y**4 / x**4
if simplified_expression == option_a:
print("Option A is correct")
else:
print("Option A is incorrect")

Option B: $\frac{7 y^3}{x^2}$

Let's compare the simplified expression with Option B.

# Define Option B
option_b = 7 * y**3 / x**2
if simplified_expression == option_b:
print("Option B is correct")
else:
print("Option B is incorrect")

Option C: $7 x^4$

Let's compare the simplified expression with Option C.

# Define Option C
option_c = 7 * x**4
if simplified_expression == option_c:
print("Option C is correct")
else:
print("Option C is incorrect")

Conclusion

In this article, we simplified the given expression $\frac{14 x^4 y^6}{7 x^6 y^2}$ and compared it with the options provided to select the correct equivalent expression. The correct equivalent expression is Option B: $\frac{7 y^3}{x^2}$ .

Final Answer

Understanding the Problem

Q: What are the rules of exponents?

A: The rules of exponents are used to simplify expressions with exponents. The main rules are:

Product of Powers Rule: When multiplying two powers with the same base, add the exponents.
Power of a Power Rule: When raising a power to another power, multiply the exponents.
Power of a Product Rule: When raising a product to a power, raise each factor to that power.
Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, follow these steps:

Combine like terms: Combine any like terms in the expression.
Apply the product of powers rule: If the expression contains a product of powers, apply the product of powers rule to simplify the expression.
Apply the power of a power rule: If the expression contains a power of a power, apply the power of a power rule to simplify the expression.
Apply the power of a product rule: If the expression contains a power of a product, apply the power of a product rule to simplify the expression.
Apply the quotient of powers rule: If the expression contains a quotient of powers, apply the quotient of powers rule to simplify the expression.

Q: How do I cancel out common factors in an expression?

A: To cancel out common factors in an expression, follow these steps:

Factor the numerator and denominator: Factor the numerator and denominator of the expression.
Cancel out common factors: Cancel out any common factors between the numerator and denominator.

Q: What is the difference between a simplified expression and an equivalent expression?

A: A simplified expression is an expression that has been simplified using the rules of exponents and other simplification techniques. An equivalent expression is an expression that has the same value as the original expression, but may not be in the simplest form.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, follow these steps:

Simplify both expressions: Simplify both expressions using the rules of exponents and other simplification techniques.
Compare the simplified expressions: Compare the simplified expressions to determine if they are equal.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Not combining like terms: Failing to combine like terms in an expression can lead to an incorrect simplified expression.
Not applying the product of powers rule: Failing to apply the product of powers rule can lead to an incorrect simplified expression.
Not applying the power of a power rule: Failing to apply the power of a power rule can lead to an incorrect simplified expression.
Not applying the power of a product rule: Failing to apply the power of a product rule can lead to an incorrect simplified expression.
Not applying the quotient of powers rule: Failing to apply the quotient of powers rule can lead to an incorrect simplified expression.

Conclusion

In this article, we discussed the rules of exponents and how to simplify expressions using these rules. We also discussed how to determine if two expressions are equivalent and some common mistakes to avoid when simplifying expressions. By following these steps and avoiding common mistakes, you can simplify expressions with confidence.

Final Answer

The final answer is $\boxed{\frac{7 y^3}{x^2}}$ .