Which Of The Following Choices Are Equivalent To The Expression Below? Check All That Apply: X 9 / 4 X^{9 / 4} X 9/4 A. X 9 4 \sqrt[4]{x^9} 4 X 9 ​ B. ( X 4 ) 1 / 9 \left(x^4\right)^{1 / 9} ( X 4 ) 1/9 C. ( X 9 ) 1 / 4 \left(x^9\right)^{1 / 4} ( X 9 ) 1/4 D. ( X 7 ) 4 (\sqrt[7]{x})^4 ( 7 X ​ ) 4

Introduction

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 equivalent expressions, specifically focusing on the given expression $x^{9 / 4}$ and its possible equivalents.

What are Exponents and Radicals?

Before we dive into the discussion, let's briefly review the concepts of exponents and radicals.

• Exponents: Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$.
• Radicals: Radicals, on the other hand, are a way of representing roots. For example, $\sqrt{x}$ means the square root of $x$.

The Given Expression: $x^{9 / 4}$

The given expression is $x^{9 / 4}$. To understand this expression, let's break it down:

• The exponent $9 / 4$ can be rewritten as $2.25$.
• This means that the expression $x^{9 / 4}$ is equivalent to $x^{2.25}$.

Option A: $\sqrt[4]{x^9}$

Let's analyze Option A: $\sqrt[4]{x^9}$.

• The radical $\sqrt[4]{x^9}$ means the fourth root of $x^9$.
• Using the property of radicals, we can rewrite this expression as $(x^9)^{1/4}$.
• This is equivalent to $x^{9/4}$, which is the given expression.

Option B: $\left(x^4\right)^{1 / 9}$

Now, let's analyze Option B: $\left(x^4\right)^{1 / 9}$.

• The expression $\left(x^4\right)^{1 / 9}$ means the ninth root of $x^4$.
• Using the property of exponents, we can rewrite this expression as $x^{4/9}$.
• This is not equivalent to the given expression $x^{9/4}$.

Option C: $\left(x^9\right)^{1 / 4}$

Next, let's analyze Option C: $\left(x^9\right)^{1 / 4}$.

• The expression $\left(x^9\right)^{1 / 4}$ means the fourth root of $x^9$.
• Using the property of radicals, we can rewrite this expression as $\sqrt[4]{x^9}$.
• This is equivalent to the given expression $x^{9/4}$.

Option D: $(\sqrt[7]{x})^4$

Finally, let's analyze Option D: $(\sqrt[7]{x})^4$.

• The expression $(\sqrt[7]{x})^4$ means the fourth power of the seventh root of $x$.
• Using the property of radicals, we can rewrite this expression as $x^{4/7}$.
• This is not equivalent to the given expression $x^{9/4}$.

Conclusion

In conclusion, the equivalent expressions to the given expression $x^{9 / 4}$ are:

• Option A: $\sqrt[4]{x^9}$
• Option C: $\left(x^9\right)^{1 / 4}$

These expressions are equivalent to the given expression $x^{9/4}$ because they all represent the same mathematical operation. The other options, B and D, are not equivalent to the given expression.

Key Takeaways

• Exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions and solve equations.
• The given expression $x^{9 / 4}$ can be rewritten as $x^{2.25}$.
• The equivalent expressions to the given expression are $\sqrt[4]{x^9}$ and $\left(x^9\right)^{1 / 4}$.

Practice Problems

To reinforce your understanding of exponents and radicals, try solving the following practice problems:

1. Simplify the expression $\left(x^3\right)^{1/2}$.
2. Rewrite the expression $\sqrt[3]{x^6}$ using exponents.
3. Simplify the expression $\left(x^4\right)^{1/3}$.

References

• Exponents and Radicals
• Equivalent Expressions

Glossary

• Exponent: A shorthand way of representing repeated multiplication.
• Radical: A way of representing roots.
• Equivalent Expressions: Expressions that represent the same mathematical operation.
  Exponents and Radicals: A Q&A Guide =====================================

Introduction

Exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. In this article, we will answer some frequently asked questions about exponents and radicals.

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

A: An exponent is a shorthand way of representing repeated multiplication, while a radical is a way of representing roots.

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

A: To simplify an expression with an exponent, you can use the following properties:

• Product of Powers: When multiplying two powers with the same base, add the exponents.
• Power of a Power: When raising a power to a power, multiply the exponents.
• Zero Exponent: Any non-zero number raised to the zero power is equal to 1.

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

A: To simplify an expression with a radical, you can use the following properties:

• Product of Radicals: When multiplying two radicals, multiply the radicands and keep the same index.
• Quotient of Radicals: When dividing two radicals, divide the radicands and keep the same index.
• Rationalizing the Denominator: To rationalize the denominator of a fraction with a radical, multiply the numerator and denominator by the conjugate of the denominator.

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

A: A positive exponent represents a power, while a negative exponent represents a reciprocal.

Q: How do I handle negative exponents?

A: To handle negative exponents, you can use the following property:

• Negative Exponent: Any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power.

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

A: A radical represents a root, while a rational exponent represents a power.

Q: How do I convert a radical to a rational exponent?

A: To convert a radical to a rational exponent, you can use the following property:

• Radical to Rational Exponent: Any radical can be rewritten as a rational exponent using the following formula: $\sqrt[n]{x} = x^{1/n}$.

Q: How do I convert a rational exponent to a radical?

A: To convert a rational exponent to a radical, you can use the following property:

• Rational Exponent to Radical: Any rational exponent can be rewritten as a radical using the following formula: $x^{m/n} = \sqrt[n]{x^m}$.

Q: What are some common mistakes to avoid when working with exponents and radicals?

A: Some common mistakes to avoid when working with exponents and radicals include:

• Forgetting to simplify expressions: Make sure to simplify expressions with exponents and radicals.
• Misusing the properties of exponents and radicals: Make sure to use the properties of exponents and radicals correctly.
• Not checking for negative exponents: Make sure to check for negative exponents and handle them correctly.

Conclusion

In conclusion, exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. By understanding the properties of exponents and radicals, you can simplify expressions and solve equations with ease.

Practice Problems

To reinforce your understanding of exponents and radicals, try solving the following practice problems:

1. Simplify the expression $\left(x^3\right)^{1/2}$.
2. Rewrite the expression $\sqrt[3]{x^6}$ using exponents.
3. Simplify the expression $\left(x^4\right)^{1/3}$.

References

• Exponents and Radicals
• Equivalent Expressions

Glossary

• Exponent: A shorthand way of representing repeated multiplication.
• Radical: A way of representing roots.
• Equivalent Expressions: Expressions that represent the same mathematical operation.