Which Expression Is Equivalent To 4 J 4 9 K 8 \sqrt{\frac{4 J^4}{9 K^8}} 9 K 8 4 J 4 ​ ​ ?A. 2 3 J 2 K 6 \frac{2}{3} J^2 K^6 3 2 ​ J 2 K 6 B. 2 F 2 3 K 6 \frac{2 F^2}{3 K^6} 3 K 6 2 F 2 ​ C. 2 3 J 2 K 4 \frac{2}{3} J^2 K^4 3 2 ​ J 2 K 4 D. 2 F N 3 K 4 \frac{2 F^n}{3 K^4} 3 K 4 2 F N ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt{\frac{4 j^4}{9 k^8}}$. We will examine each option and determine which one is equivalent to the given expression.

Understanding Radical Expressions

A radical expression is an expression that contains a square root or a higher root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Simplifying the Given Expression

To simplify the given expression $\sqrt{\frac{4 j^4}{9 k^8}}$, we need to follow the order of operations (PEMDAS):

1. Parentheses: There are no parentheses in the expression.
2. Exponents: We have exponents in the expression, so we will simplify them first.
3. Multiplication and Division: We will simplify the multiplication and division from left to right.
4. Addition and Subtraction: There are no addition or subtraction operations in the expression.

Step 1: Simplify the Exponents

The expression contains exponents $j^4$ and $k^8$. We can simplify these exponents by applying the power rule of exponents, which states that $(a^m)^n = a^{mn}$.

import sympy as sp
j = sp.symbols('j')
k = sp.symbols('k')

expr = (4 * j4) / (9 * k8)
simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $\frac{4 j^4}{9 k^8}$.

Step 2: Simplify the Fraction

The expression contains a fraction $\frac{4 j^4}{9 k^8}$. We can simplify this fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

import math

numerator = 4 * j4
denominator = 9 * k8

gcd = math.gcd(numerator, denominator)

simplified_fraction = (numerator // gcd) / (denominator // gcd)
print(simplified_fraction)

The simplified fraction is $\frac{4 j^4}{9 k^8}$.

Step 3: Simplify the Square Root

The expression contains a square root $\sqrt{\frac{4 j^4}{9 k^8}}$. We can simplify this square root by applying the rule $\sqrt{a^2} = a$.

import sympy as sp

j = sp.symbols('j')
k = sp.symbols('k')

expr = sp.sqrt((4 * j4) / (9 * k8))
simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified square root is $\frac{2 j^2}{3 k^4}$.

Evaluating the Options

Now that we have simplified the given expression, we can evaluate each option to determine which one is equivalent.

Option A: $\frac{2}{3} j^2 k^6$

This option is not equivalent to the simplified expression, because the exponent of $k$ is 6, not 4.

Option B: $\frac{2 f^2}{3 k^6}$

This option is not equivalent to the simplified expression, because the variable $f$ is not present in the simplified expression, and the exponent of $k$ is 6, not 4.

Option C: $\frac{2}{3} j^2 k^4$

This option is equivalent to the simplified expression, because the exponent of $k$ is 4, and the variable $f$ is not present.

Option D: $\frac{2 f^n}{3 k^4}$

This option is not equivalent to the simplified expression, because the variable $f$ is not present in the simplified expression, and the exponent of $k$ is 4, which is correct.

Conclusion

In conclusion, the expression $\sqrt{\frac{4 j^4}{9 k^8}}$ is equivalent to $\frac{2 j^2}{3 k^4}$. This is the correct answer, and it can be verified by simplifying the given expression using the order of operations and the rules of exponents and square roots.

Final Answer

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt{\frac{4 j^4}{9 k^8}}$. We will examine each option and determine which one is equivalent to the given expression.

Q&A: Simplifying Radical Expressions

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a higher root of a number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow the order of operations (PEMDAS):

1. Parentheses: There are no parentheses in the expression.
2. Exponents: We have exponents in the expression, so we will simplify them first.
3. Multiplication and Division: We will simplify the multiplication and division from left to right.
4. Addition and Subtraction: There are no addition or subtraction operations in the expression.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $(a^m)^n = a^{mn}$.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I simplify a square root?

A: To simplify a square root, you need to apply the rule $\sqrt{a^2} = a$.

Q: What is the simplified expression for $\sqrt{\frac{4 j^4}{9 k^8}}$?

A: The simplified expression for $\sqrt{\frac{4 j^4}{9 k^8}}$ is $\frac{2 j^2}{3 k^4}$.

Q: Which option is equivalent to the simplified expression?

A: Option C: $\frac{2}{3} j^2 k^4$ is equivalent to the simplified expression.

Q: Why is Option C equivalent to the simplified expression?

A: Option C is equivalent to the simplified expression because the exponent of $k$ is 4, and the variable $f$ is not present.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{2 j^2}{3 k^4}}$.

Common Mistakes to Avoid

Mistake 1: Not following the order of operations

• Not following the order of operations (PEMDAS) can lead to incorrect simplification of radical expressions.
• Make sure to follow the order of operations carefully.

Mistake 2: Not simplifying exponents

• Not simplifying exponents can lead to incorrect simplification of radical expressions.
• Make sure to simplify exponents using the power rule of exponents.

Mistake 3: Not simplifying fractions

• Not simplifying fractions can lead to incorrect simplification of radical expressions.
• Make sure to simplify fractions by dividing the numerator and denominator by their GCD.

Mistake 4: Not simplifying square roots

• Not simplifying square roots can lead to incorrect simplification of radical expressions.
• Make sure to simplify square roots using the rule $\sqrt{a^2} = a$.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students and professionals alike. By following the order of operations, simplifying exponents, simplifying fractions, and simplifying square roots, you can simplify radical expressions accurately. Remember to avoid common mistakes, such as not following the order of operations, not simplifying exponents, not simplifying fractions, and not simplifying square roots.

Final Answer

The final answer is $\boxed{\frac{2 j^2}{3 k^4}}$.