Which Expression Is Equivalent To $\sqrt[3]{\frac{10 X^5}{54 X^8}}$? Assume $x \neq 0$.A. $\frac{\sqrt[3]{10 X}}{3 X^2}$B. $\frac{x(\sqrt[3]{5 X})}{3}$C. $\frac{3(\sqrt[3]{5 X})}{x}$D.

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Introduction

Radical expressions, also known as roots, are a fundamental concept in mathematics. They are used to represent the nth root of a number, where n is a positive integer. In this article, we will explore the concept of simplifying radical expressions, with a focus on the given problem: 10x554x83\sqrt[3]{\frac{10 x^5}{54 x^8}}. We will break down the solution step by step, using the properties of exponents and radicals to simplify the expression.

Understanding the Problem

The given problem is to simplify the expression 10x554x83\sqrt[3]{\frac{10 x^5}{54 x^8}}. To start, we need to understand the properties of exponents and radicals. The expression inside the cube root is a fraction, with the numerator being 10x510 x^5 and the denominator being 54x854 x^8.

Step 1: Simplify the Fraction

To simplify the fraction, we can start by factoring out the greatest common factor (GCF) of the numerator and denominator. In this case, the GCF is 2x52 x^5. We can rewrite the fraction as:

10x554x8=5x27x3\frac{10 x^5}{54 x^8} = \frac{5 x}{27 x^3}

Step 2: Apply the Property of Cube Roots

Now that we have simplified the fraction, we can apply the property of cube roots. The cube root of a fraction can be rewritten as the cube root of the numerator divided by the cube root of the denominator. In this case, we have:

5x27x33=5x327x33\sqrt[3]{\frac{5 x}{27 x^3}} = \frac{\sqrt[3]{5 x}}{\sqrt[3]{27 x^3}}

Step 3: Simplify the Cube Roots

Now that we have applied the property of cube roots, we can simplify the cube roots. The cube root of 27x327 x^3 can be rewritten as 3x3 x. We can rewrite the expression as:

5x327x33=5x33x\frac{\sqrt[3]{5 x}}{\sqrt[3]{27 x^3}} = \frac{\sqrt[3]{5 x}}{3 x}

Step 4: Simplify the Expression

Now that we have simplified the cube roots, we can simplify the expression further. We can rewrite the expression as:

5x33x=x(5x3)3x2\frac{\sqrt[3]{5 x}}{3 x} = \frac{x(\sqrt[3]{5 x})}{3 x^2}

Conclusion

In conclusion, the expression 10x554x83\sqrt[3]{\frac{10 x^5}{54 x^8}} is equivalent to x(5x3)3x2\frac{x(\sqrt[3]{5 x})}{3 x^2}. We simplified the expression by factoring out the GCF, applying the property of cube roots, and simplifying the cube roots. The final answer is:

The Final Answer

The final answer is x(5x3)3x2\boxed{\frac{x(\sqrt[3]{5 x})}{3 x^2}}.

Discussion

This problem requires a good understanding of the properties of exponents and radicals. The student needs to be able to simplify the fraction, apply the property of cube roots, and simplify the cube roots. The student also needs to be able to rewrite the expression in a simpler form.

Tips and Tricks

  • Make sure to factor out the GCF of the numerator and denominator.
  • Apply the property of cube roots to simplify the expression.
  • Simplify the cube roots by rewriting them as a product of a number and a variable.
  • Rewrite the expression in a simpler form by combining like terms.

Common Mistakes

  • Failing to factor out the GCF of the numerator and denominator.
  • Not applying the property of cube roots.
  • Not simplifying the cube roots.
  • Not rewriting the expression in a simpler form.

Real-World Applications

Radical expressions are used in many real-world applications, such as:

  • Physics: to calculate the distance traveled by an object.
  • Engineering: to calculate the stress on a material.
  • Computer Science: to calculate the time complexity of an algorithm.

Conclusion

Introduction

Radical expressions, also known as roots, are a fundamental concept in mathematics. They are used to represent the nth root of a number, where n is a positive integer. In this article, we will explore the concept of simplifying radical expressions, with a focus on the given problem: 10x554x83\sqrt[3]{\frac{10 x^5}{54 x^8}}. We will break down the solution step by step, using the properties of exponents and radicals to simplify the expression.

Q&A

Q: What is a radical expression?

A: A radical expression is a mathematical expression that involves a root, such as a square root or a cube root.

Q: What is the difference between a square root and a cube root?

A: A square root is the nth root of a number, where n is 2, while a cube root is the nth root of a number, where n is 3.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

  1. Factor out the greatest common factor (GCF) of the numerator and denominator.
  2. Apply the property of cube roots to simplify the expression.
  3. Simplify the cube roots by rewriting them as a product of a number and a variable.
  4. Rewrite the expression in a simpler form by combining like terms.

Q: What is the property of cube roots?

A: The property of cube roots states that the cube root of a fraction can be rewritten as the cube root of the numerator divided by the cube root of the denominator.

Q: How do I apply the property of cube roots?

A: To apply the property of cube roots, you need to rewrite the expression as the cube root of the numerator divided by the cube root of the denominator.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest number that divides both the numerator and denominator of a fraction.

Q: How do I find the GCF?

A: To find the GCF, you need to list the factors of the numerator and denominator and find the largest number that is common to both.

Q: What is the difference between a like term and an unlike term?

A: A like term is a term that has the same variable and exponent, while an unlike term is a term that has a different variable or exponent.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Common Mistakes

  • Failing to factor out the GCF of the numerator and denominator.
  • Not applying the property of cube roots.
  • Not simplifying the cube roots.
  • Not rewriting the expression in a simpler form.

Real-World Applications

Radical expressions are used in many real-world applications, such as:

  • Physics: to calculate the distance traveled by an object.
  • Engineering: to calculate the stress on a material.
  • Computer Science: to calculate the time complexity of an algorithm.

Conclusion

In conclusion, simplifying radical expressions is an important concept in mathematics. The student needs to be able to simplify the fraction, apply the property of cube roots, and simplify the cube roots. The student also needs to be able to rewrite the expression in a simpler form. With practice and patience, the student can master this concept and apply it to real-world problems.

Practice Problems

  1. Simplify the expression 12x436x73\sqrt[3]{\frac{12 x^4}{36 x^7}}.
  2. Simplify the expression 20x380x63\sqrt[3]{\frac{20 x^3}{80 x^6}}.
  3. Simplify the expression 15x245x53\sqrt[3]{\frac{15 x^2}{45 x^5}}.

Answer Key

  1. x(3x3)3x2\frac{x(\sqrt[3]{3 x})}{3 x^2}
  2. x(5x3)3x2\frac{x(\sqrt[3]{5 x})}{3 x^2}
  3. x(3x3)3x2\frac{x(\sqrt[3]{3 x})}{3 x^2}