Simplify The Expression:$\[ 5 \sqrt[3]{432 A^5 B^6} \\]

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Understanding the Problem

When dealing with radical expressions, it's essential to simplify them by factoring out perfect cubes from the radicand. This process involves breaking down the radicand into its prime factors and identifying the perfect cubes that can be extracted from the expression. In this case, we're given the expression $5 \sqrt[3]{432 a^5 b^6}$, and our goal is to simplify it.

Breaking Down the Radicand

To simplify the expression, we need to start by breaking down the radicand, which is $432 a^5 b^6$. We can begin by factoring out the largest perfect cube from the radicand. To do this, we need to identify the prime factors of $432$ and $a^5 b^6$.

Prime Factorization of 432

The prime factorization of $432$ is $2^4 \cdot 3^3$. This means that $432$ can be expressed as the product of $2^4$ and $3^3$.

Prime Factorization of $a^5 b^6$

The prime factorization of $a^5 b^6$ is $a^3 a^2 b^3 b^3$. This means that $a^5 b^6$ can be expressed as the product of $a^3$, $a^2$, $b^3$, and $b^3$.

Combining the Prime Factorizations

Now that we have the prime factorizations of $432$ and $a^5 b^6$, we can combine them to get the prime factorization of the radicand. This gives us:

$$432 a^5 b^6 = 2^4 \cdot 3^3 \cdot a^3 a^2 b^3 b^3$$

Identifying Perfect Cubes

Now that we have the prime factorization of the radicand, we can identify the perfect cubes that can be extracted from the expression. We can see that $3^3$ is a perfect cube, and we can also see that $a^3$ is a perfect cube.

Simplifying the Expression

Now that we have identified the perfect cubes, we can simplify the expression by extracting them from the radicand. This gives us:

$$5 \sqrt[3]{432 a^5 b^6} = 5 \sqrt[3]{2^4 \cdot 3^3 \cdot a^3 a^2 b^3 b^3}$$

Extracting Perfect Cubes

We can now extract the perfect cubes from the radicand. This gives us:

$$5 \sqrt[3]{2^4 \cdot 3^3 \cdot a^3 a^2 b^3 b^3} = 5 \cdot 2 \cdot 3 \cdot a \cdot \sqrt[3]{2^2 a^2 b^3}$$

Final Simplification

Now that we have extracted the perfect cubes, we can simplify the expression further. This gives us:

$$5 \cdot 2 \cdot 3 \cdot a \cdot \sqrt[3]{2^2 a^2 b^3} = 30 a \sqrt[3]{4 a^2 b^3}$$

Conclusion

In this article, we simplified the expression $5 \sqrt[3]{432 a^5 b^6}$ by factoring out perfect cubes from the radicand. We broke down the radicand into its prime factors, identified the perfect cubes, and extracted them from the radicand. This process involved understanding the prime factorization of the radicand, identifying perfect cubes, and simplifying the expression. The final simplified expression is $30 a \sqrt[3]{4 a^2 b^3}$.

Frequently Asked Questions

• Q: What is the prime factorization of $432$? A: The prime factorization of $432$ is $2^4 \cdot 3^3$.
• Q: What is the prime factorization of $a^5 b^6$? A: The prime factorization of $a^5 b^6$ is $a^3 a^2 b^3 b^3$.
• Q: How do we simplify the expression $5 \sqrt[3]{432 a^5 b^6}$? A: We simplify the expression by factoring out perfect cubes from the radicand, breaking down the radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Key Takeaways

• To simplify a radical expression, we need to factor out perfect cubes from the radicand.
• We can break down the radicand into its prime factors to identify the perfect cubes.
• We can extract the perfect cubes from the radicand to simplify the expression.
• The final simplified expression is $30 a \sqrt[3]{4 a^2 b^3}$.

Further Reading

• Radical Expressions: A Comprehensive Guide
• Simplifying Radical Expressions: A Step-by-Step Guide
• Perfect Cubes: A Guide to Factoring and Simplifying
  

Introduction

Radical expressions can be complex and challenging to simplify. However, with the right techniques and strategies, you can simplify even the most difficult expressions. In this article, we'll answer some of the most frequently asked questions about simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a radical sign, which is denoted by the symbol $\sqrt[n]{x}$. The radical sign indicates that the expression inside the sign is to be taken as the nth root of the expression.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is an expression that contains a radical sign, while an exponential expression is an expression that contains an exponent. For example, $\sqrt[3]{x}$ is a radical expression, while $x^3$ is an exponential expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to factor out perfect cubes from the radicand. This involves breaking down the radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Q: What is a perfect cube?

A: A perfect cube is a number or expression that can be expressed as the cube of an integer. For example, $8$ is a perfect cube because it can be expressed as $2^3$, and $27$ is a perfect cube because it can be expressed as $3^3$.

Q: How do I identify perfect cubes in a radical expression?

A: To identify perfect cubes in a radical expression, you need to break down the radicand into its prime factors. Look for groups of three identical factors, and these will be the perfect cubes.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent. To do this, you need to take the reciprocal of the expression and change the sign of the exponent.

Q: How do I simplify a radical expression with a variable in the radicand?

A: To simplify a radical expression with a variable in the radicand, you need to factor out perfect cubes from the radicand. This involves breaking down the radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Q: Can I simplify a radical expression with a fraction in the radicand?

A: Yes, you can simplify a radical expression with a fraction in the radicand. To do this, you need to simplify the fraction and then factor out perfect cubes from the radicand.

Q: How do I simplify a radical expression with multiple radicals?

A: To simplify a radical expression with multiple radicals, you need to factor out perfect cubes from each radicand. This involves breaking down each radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

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

A: Yes, you can simplify a radical expression with a radical in the denominator. To do this, you need to rationalize the denominator by multiplying the expression by a clever form of 1.

Q: How do I simplify a radical expression with a complex number in the radicand?

A: To simplify a radical expression with a complex number in the radicand, you need to factor out perfect cubes from the radicand. This involves breaking down the radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Q: Can I simplify a radical expression with a radical in the numerator and denominator?

A: Yes, you can simplify a radical expression with a radical in the numerator and denominator. To do this, you need to rationalize the denominator by multiplying the expression by a clever form of 1.

Q: How do I simplify a radical expression with a variable in the denominator?

A: To simplify a radical expression with a variable in the denominator, you need to factor out perfect cubes from the radicand. This involves breaking down the radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Q: Can I simplify a radical expression with a fraction in the denominator?

A: Yes, you can simplify a radical expression with a fraction in the denominator. To do this, you need to simplify the fraction and then factor out perfect cubes from the radicand.

Q: How do I simplify a radical expression with multiple variables in the radicand?

A: To simplify a radical expression with multiple variables in the radicand, you need to factor out perfect cubes from each radicand. This involves breaking down each radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Q: Can I simplify a radical expression with a radical in the numerator and a fraction in the denominator?

A: Yes, you can simplify a radical expression with a radical in the numerator and a fraction in the denominator. To do this, you need to rationalize the denominator by multiplying the expression by a clever form of 1.

Q: How do I simplify a radical expression with a complex number in the numerator and a fraction in the denominator?

A: To simplify a radical expression with a complex number in the numerator and a fraction in the denominator, you need to factor out perfect cubes from the radicand. This involves breaking down the radicand into its prime factors, identifying perfect cubes, and extracting them from the radicand.

Q: Can I simplify a radical expression with a radical in the numerator and a complex number in the denominator?

A: Yes, you can simplify a radical expression with a radical in the numerator and a complex number in the denominator. To do this, you need to rationalize the denominator by multiplying the expression by a clever form of 1.

Conclusion

Simplifying radical expressions can be challenging, but with the right techniques and strategies, you can simplify even the most difficult expressions. By understanding the concepts of perfect cubes, prime factorization, and rationalizing the denominator, you can simplify radical expressions with ease. Remember to always factor out perfect cubes from the radicand, break down the radicand into its prime factors, and extract them from the radicand to simplify the expression.

Frequently Asked Questions

• Q: What is a radical expression? A: A radical expression is an expression that contains a radical sign, which is denoted by the symbol $\sqrt[n]{x}$.
• Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to factor out perfect cubes from the radicand.
• Q: What is a perfect cube? A: A perfect cube is a number or expression that can be expressed as the cube of an integer.
• Q: How do I identify perfect cubes in a radical expression? A: To identify perfect cubes in a radical expression, you need to break down the radicand into its prime factors.

Key Takeaways

• To simplify a radical expression, you need to factor out perfect cubes from the radicand.
• You can break down the radicand into its prime factors to identify perfect cubes.
• You can extract perfect cubes from the radicand to simplify the expression.
• The final simplified expression is $30 a \sqrt[3]{4 a^2 b^3}$.

Further Reading

• Radical Expressions: A Comprehensive Guide
• Simplifying Radical Expressions: A Step-by-Step Guide
• Perfect Cubes: A Guide to Factoring and Simplifying