Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Express in Simplest Radical Form
Given expression:
\[ 4x^{\frac{3}{2}} \]
To express this in its simplest radical form, follow these steps:
1. **Understand the Fractional Exponent:** The exponent \(\frac{3}{2}\) consists of a numerator (3) and a denominator (2). The denominator indicates the root (in this case, a square root), and the numerator indicates the power.
2. **Rewrite the Expression:** The exponent \(\frac{3}{2}\) can be rewritten as a radical.
\[ x^{\frac{3}{2}} = (x^{3})^{\frac{1}{2}} = \sqrt{x^3} \]
3. **Combine with Coefficient:** Apply the coefficient 4 to the simplified radical form.
\[ 4x^{\frac{3}{2}} = 4 \sqrt{x^3} \]
4. **Simplify the Radical:** If possible, simplify the radical further. For \(\sqrt{x^3}\):
\[ \sqrt{x^3} = \sqrt{x^2 \cdot x} = x \sqrt{x} \]
5. **Final Simplified Form:** Combine all parts to get the final expression in simplest radical form.
\[ 4x \sqrt{x} \]
Therefore, the expression \( 4x^{\frac{3}{2}} \) in its simplest radical form is:
\[ 4x \sqrt{x} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd778ba75-86ed-4d38-af28-a715226ea6d4%2F0cb37153-3b84-4b3a-9fbd-14b8749981de%2Fht23hva_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Express in Simplest Radical Form
Given expression:
\[ 4x^{\frac{3}{2}} \]
To express this in its simplest radical form, follow these steps:
1. **Understand the Fractional Exponent:** The exponent \(\frac{3}{2}\) consists of a numerator (3) and a denominator (2). The denominator indicates the root (in this case, a square root), and the numerator indicates the power.
2. **Rewrite the Expression:** The exponent \(\frac{3}{2}\) can be rewritten as a radical.
\[ x^{\frac{3}{2}} = (x^{3})^{\frac{1}{2}} = \sqrt{x^3} \]
3. **Combine with Coefficient:** Apply the coefficient 4 to the simplified radical form.
\[ 4x^{\frac{3}{2}} = 4 \sqrt{x^3} \]
4. **Simplify the Radical:** If possible, simplify the radical further. For \(\sqrt{x^3}\):
\[ \sqrt{x^3} = \sqrt{x^2 \cdot x} = x \sqrt{x} \]
5. **Final Simplified Form:** Combine all parts to get the final expression in simplest radical form.
\[ 4x \sqrt{x} \]
Therefore, the expression \( 4x^{\frac{3}{2}} \) in its simplest radical form is:
\[ 4x \sqrt{x} \]
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