Complete The Steps To Simplify 4 5 ⋅ 2 \sqrt[5]{4} \cdot \sqrt{2} 5 4 ​ ⋅ 2 ​ .Rewrite Using Rational Exponents:A. 2 2 5 ⋅ 2 1 2 2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}} 2 5 2 ​ ⋅ 2 2 1 ​ B. 4 5 ⋅ 2 2 4^5 \cdot 2^2 4 5 ⋅ 2 2 C. 4 1 5 ⋅ 2 1 7 4^{\frac{1}{5}} \cdot 2^{\frac{1}{7}} 4 5 1 ​ ⋅ 2 7 1 ​

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Radical expressions can be simplified using various techniques, including rational exponents. In this article, we will explore the steps to simplify the expression $\sqrt[5]{4} \cdot \sqrt{2}$ using rational exponents.

Understanding Rational Exponents

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Rational exponents are a way to express roots using fractional exponents. For example, $\sqrt[n]{a}$ can be written as $a^{\frac{1}{n}}$. This allows us to simplify radical expressions by combining them with other numbers.

Step 1: Rewrite the Radical Expressions Using Rational Exponents

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To simplify the expression $\sqrt[5]{4} \cdot \sqrt{2}$, we need to rewrite each radical expression using rational exponents.

• $\sqrt[5]{4}$ can be written as $4^{\frac{1}{5}}$
• $\sqrt{2}$ can be written as $2^{\frac{1}{2}}$

Step 2: Apply the Product Rule for Exponents

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When multiplying two numbers with the same base, we can add their exponents. In this case, we have:

$4^{\frac{1}{5}} \cdot 2^{\frac{1}{2}}$

Using the product rule for exponents, we can rewrite this expression as:

$(4 \cdot 2)^{\frac{1}{5} + \frac{1}{2}}$

Step 3: Simplify the Expression Inside the Parentheses

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To simplify the expression inside the parentheses, we need to multiply the numbers and add their exponents.

$4 \cdot 2 = 8$

$\frac{1}{5} + \frac{1}{2} = \frac{2}{10} + \frac{5}{10} = \frac{7}{10}$

So, the expression becomes:

$8^{\frac{7}{10}}$

Step 4: Rewrite the Expression Using a Rational Exponent

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We can rewrite the expression $8^{\frac{7}{10}}$ using a rational exponent as:

$2^{\frac{14}{10}}$

Step 5: Simplify the Rational Exponent

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To simplify the rational exponent, we can divide the numerator and denominator by their greatest common divisor, which is 2.

$\frac{14}{10} = \frac{7}{5}$

So, the expression becomes:

$2^{\frac{7}{5}}$

Conclusion

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In this article, we simplified the expression $\sqrt[5]{4} \cdot \sqrt{2}$ using rational exponents. We rewrote each radical expression using rational exponents, applied the product rule for exponents, simplified the expression inside the parentheses, and finally rewrote the expression using a rational exponent.

The final answer is $2^{\frac{7}{5}}$.

Answer Choices

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A. $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$ B. $4^5 \cdot 2^2$ C. $4^{\frac{1}{5}} \cdot 2^{\frac{1}{7}}$

Correct Answer

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A. $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$

Explanation

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The correct answer is A. $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$. This is because we simplified the expression $\sqrt[5]{4} \cdot \sqrt{2}$ using rational exponents, and the final answer is $2^{\frac{7}{5}}$, which can be rewritten as $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$.

Discussion

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Radical expressions can be simplified using various techniques, including rational exponents. In this article, we explored the steps to simplify the expression $\sqrt[5]{4} \cdot \sqrt{2}$ using rational exponents. We rewrote each radical expression using rational exponents, applied the product rule for exponents, simplified the expression inside the parentheses, and finally rewrote the expression using a rational exponent.

The final answer is $2^{\frac{7}{5}}$, which can be rewritten as $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$. This is the correct answer, and it demonstrates the power of using rational exponents to simplify radical expressions.

Related Topics

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• Simplifying radical expressions using rational exponents
• Applying the product rule for exponents
• Simplifying expressions inside parentheses
• Rewriting expressions using rational exponents

Further Reading

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• Simplifying Radical Expressions
• Rational Exponents
• Product Rule for Exponents

References

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• Math Is Fun
• Khan Academy
• Mathway
  

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Radical expressions can be simplified using various techniques, including rational exponents. In this article, we will explore some common questions and answers related to simplifying radical expressions.

Q: What is a radical expression?

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A: A radical expression is an expression that contains a root, such as a square root, cube root, or higher root.

Q: What is a rational exponent?

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A: A rational exponent is a way to express a root using a fractional exponent. For example, $\sqrt[n]{a}$ can be written as $a^{\frac{1}{n}}$.

Q: How do I simplify a radical expression using rational exponents?

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A: To simplify a radical expression using rational exponents, follow these steps:

1. Rewrite each radical expression using rational exponents.
2. Apply the product rule for exponents.
3. Simplify the expression inside the parentheses.
4. Rewrite the expression using a rational exponent.

Q: What is the product rule for exponents?

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A: The product rule for exponents states that when multiplying two numbers with the same base, we can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do I simplify an expression inside parentheses?

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A: To simplify an expression inside parentheses, follow these steps:

1. Multiply the numbers inside the parentheses.
2. Add the exponents of the numbers inside the parentheses.

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

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A: A rational exponent is a way to express a root using a fractional exponent, while a fractional exponent is a way to express a power using a fractional exponent. For example, $\sqrt[n]{a}$ can be written as $a^{\frac{1}{n}}$, while $a^{\frac{1}{n}}$ can be written as $(a^{\frac{1}{n}})^n$.

Q: Can I simplify a radical expression using rational exponents if it contains multiple roots?

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A: Yes, you can simplify a radical expression using rational exponents if it contains multiple roots. For example, $\sqrt[5]{4} \cdot \sqrt{2}$ can be simplified using rational exponents as follows:

$4^{\frac{1}{5}} \cdot 2^{\frac{1}{2}}$

Using the product rule for exponents, we can rewrite this expression as:

$(4 \cdot 2)^{\frac{1}{5} + \frac{1}{2}}$

Simplifying the expression inside the parentheses, we get:

$8^{\frac{7}{10}}$

Rewriting the expression using a rational exponent, we get:

$2^{\frac{14}{10}}$

Simplifying the rational exponent, we get:

$2^{\frac{7}{5}}$

Q: Can I simplify a radical expression using rational exponents if it contains a negative exponent?

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A: Yes, you can simplify a radical expression using rational exponents if it contains a negative exponent. For example, $\sqrt[n]{\frac{1}{a}}$ can be simplified using rational exponents as follows:

$\frac{1}{a^{\frac{1}{n}}}$

Using the product rule for exponents, we can rewrite this expression as:

$a^{-\frac{1}{n}}$

Rewriting the expression using a rational exponent, we get:

$a^{\frac{-1}{n}}$

Q: What are some common mistakes to avoid when simplifying radical expressions using rational exponents?

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A: Some common mistakes to avoid when simplifying radical expressions using rational exponents include:

• Not rewriting each radical expression using rational exponents.
• Not applying the product rule for exponents.
• Not simplifying the expression inside the parentheses.
• Not rewriting the expression using a rational exponent.

Q: How do I check my work when simplifying radical expressions using rational exponents?

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A: To check your work when simplifying radical expressions using rational exponents, follow these steps:

1. Rewrite each radical expression using rational exponents.
2. Apply the product rule for exponents.
3. Simplify the expression inside the parentheses.
4. Rewrite the expression using a rational exponent.
5. Check that the final answer is in the correct form.

Answer Choices

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A. $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$ B. $4^5 \cdot 2^2$ C. $4^{\frac{1}{5}} \cdot 2^{\frac{1}{7}}$

Correct Answer

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A. $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$

Explanation

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The correct answer is A. $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$. This is because we simplified the expression $\sqrt[5]{4} \cdot \sqrt{2}$ using rational exponents, and the final answer is $2^{\frac{7}{5}}$, which can be rewritten as $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$.

Discussion

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Radical expressions can be simplified using various techniques, including rational exponents. In this article, we explored some common questions and answers related to simplifying radical expressions. We discussed the product rule for exponents, simplifying expressions inside parentheses, and rewriting expressions using rational exponents.

The final answer is $2^{\frac{7}{5}}$, which can be rewritten as $2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$. This is the correct answer, and it demonstrates the power of using rational exponents to simplify radical expressions.

Related Topics

---

• Simplifying radical expressions using rational exponents
• Applying the product rule for exponents
• Simplifying expressions inside parentheses
• Rewriting expressions using rational exponents

Further Reading

---

• Simplifying Radical Expressions
• Rational Exponents
• Product Rule for Exponents

References

---

• Math Is Fun
• Khan Academy
• Mathway