Which Expression Is Equivalent To $(3 Y)^{\frac{1}{6}}$?A. $\sqrt{y}$B. $\frac{3 Y}{\sqrt[6]{3 Y}}$C. $ 3 Y 6 \sqrt[6]{3 Y} 6 3 Y ​ [/tex]D. $\frac{\sqrt[6]{y}}{2}$

by ADMIN 173 views

Introduction to Exponents and Radicals

Exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. In this article, we will explore the concept of exponents and radicals, and specifically, we will examine the expression $(3 y)^{\frac{1}{6}}$ and determine which of the given options is equivalent to it.

What are Exponents and Radicals?

Exponents and radicals are two sides of the same coin. Exponents represent repeated multiplication, while radicals represent the inverse operation of exponents. In other words, exponents tell us how many times a number is multiplied by itself, while radicals tell us the number that is being multiplied by itself.

For example, the expression $(3 y)^{\frac{1}{6}}$ can be read as "3y to the power of 1/6". This means that 3y is being multiplied by itself 1/6 times.

Understanding the Expression $(3 y)^{\frac{1}{6}}$

The expression $(3 y)^{\frac{1}{6}}$ can be broken down into two parts: the base (3y) and the exponent (1/6). The base is the number that is being multiplied by itself, and the exponent is the number that tells us how many times the base is being multiplied by itself.

In this case, the base is 3y, and the exponent is 1/6. This means that 3y is being multiplied by itself 1/6 times.

Simplifying the Expression $(3 y)^{\frac{1}{6}}$

To simplify the expression $(3 y)^{\frac{1}{6}}$, we can use the rule of exponents that states (am)n=amn(a^m)^n = a^{mn}. In this case, we can rewrite the expression as $(3 y)^{\frac{1}{6}} = (3{\frac{1}{6}})(y{\frac{1}{6}})$.

Evaluating the Options

Now that we have simplified the expression $(3 y)^{\frac{1}{6}}$, we can evaluate the options to determine which one is equivalent to it.

Option A: $\sqrt{y}$

This option is not equivalent to $(3 y)^{\frac{1}{6}}$ because it only takes into account the y term and ignores the 3 term.

Option B: $\frac{3 y}{\sqrt[6]{3 y}}$

This option is not equivalent to $(3 y)^{\frac{1}{6}}$ because it divides the 3y term by the 6th root of 3y, which is not the same as multiplying 3y by itself 1/6 times.

Option C: $\sqrt[6]{3 y}$

This option is equivalent to $(3 y)^{\frac{1}{6}}$ because it takes into account both the 3 term and the y term and multiplies them by themselves 1/6 times.

Option D: $\frac{\sqrt[6]{y}}{2}$

This option is not equivalent to $(3 y)^{\frac{1}{6}}$ because it only takes into account the y term and divides it by 2, which is not the same as multiplying 3y by itself 1/6 times.

Conclusion

In conclusion, the expression $(3 y)^\frac{1}{6}}$ is equivalent to option C $\sqrt[6]{3 y$. This option takes into account both the 3 term and the y term and multiplies them by themselves 1/6 times, which is the same as the original expression.

Final Answer

The final answer is option C: $\sqrt[6]{3 y}$.

Introduction

In our previous article, we explored the concept of exponents and radicals and examined the expression $(3 y)^{\frac{1}{6}}$ to determine which of the given options is equivalent to it. In this article, we will answer some of the most frequently asked questions related to exponents and radicals.

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

A: An exponent represents repeated multiplication, while a radical represents the inverse operation of exponents. In other words, exponents tell us how many times a number is multiplied by itself, while radicals tell us the number that is being multiplied by itself.

Q: How do I simplify an expression with an exponent?

A: To simplify an expression with an exponent, you can use the rule of exponents that states (am)n=amn(a^m)^n = a^{mn}. This means that you can rewrite the expression as $(am)n = a^{mn}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent tells us how many times a number is multiplied by itself, while a negative exponent tells us how many times the reciprocal of the number is multiplied by itself. In other words, a positive exponent is the same as the number being multiplied by itself, while a negative exponent is the same as the reciprocal of the number being multiplied by itself.

Q: How do I evaluate an expression with a radical?

A: To evaluate an expression with a radical, you can use the rule of radicals that states ann=a\sqrt[n]{a^n} = a. This means that you can rewrite the expression as ann=a\sqrt[n]{a^n} = a.

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

A: A square root is a radical with an exponent of 1/2, while a cube root is a radical with an exponent of 1/3. In other words, a square root is the same as the number being multiplied by itself 1/2 times, while a cube root is the same as the number being multiplied by itself 1/3 times.

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

A: To simplify an expression with multiple exponents, you can use the rule of exponents that states (am)n=amn(a^m)^n = a^{mn}. This means that you can rewrite the expression as $(am)n = a^{mn}$.

Q: What is the difference between an exponent and a power?

A: An exponent represents repeated multiplication, while a power represents a quantity that is being multiplied by itself. In other words, an exponent is the same as the number being multiplied by itself, while a power is the same as the quantity being multiplied by itself.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you can use the rule of exponents that states a−n=1ana^{-n} = \frac{1}{a^n}. This means that you can rewrite the expression as a−n=1ana^{-n} = \frac{1}{a^n}.

Conclusion

In conclusion, exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. By understanding the rules of exponents and radicals, we can evaluate expressions and simplify complex equations.

Final Answer

The final answer is that exponents and radicals are essential concepts in mathematics that help us simplify complex expressions and solve equations.

Additional Resources

  • For more information on exponents and radicals, please refer to the following resources:
  • Khan Academy: Exponents and Radicals
  • Mathway: Exponents and Radicals
  • Wolfram Alpha: Exponents and Radicals

Common Mistakes

  • Not understanding the difference between an exponent and a radical
  • Not using the correct rules of exponents and radicals
  • Not simplifying expressions correctly

Tips and Tricks

  • Use the rule of exponents that states (am)n=amn(a^m)^n = a^{mn} to simplify expressions with multiple exponents.
  • Use the rule of radicals that states ann=a\sqrt[n]{a^n} = a to evaluate expressions with radicals.
  • Use the rule of exponents that states a−n=1ana^{-n} = \frac{1}{a^n} to evaluate expressions with negative exponents.