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Chapter 0: Problem 74

Simplify the radical expressions if possible. $$\frac{\sqrt[4]{162 x^{5}}}{\sqrt[4]{2 x}}$$

Short Answer

Expert verified
The simplified form of the expression is \(3x\).

Step by step solution

01

Simplify Numerical Part of the Radical

We first deal with the numeric part of the expression under the fourth root. In the numerator, divide 162 by 2, getting 81. The expression becomes: \(\frac{\sqrt[4]{81x^{5}}}{\sqrt[4]{x}}\).
02

Simplify Algebric Part of the Radical

Next, we simplify the algebraic part of the expression under the fourth root. Since \(x^{5}\) is divided by \(x\), we adjust the powers and get \(x^{4}\). So, the expression now becomes \(\frac{\sqrt[4]{81x^{4}}}{1}\).
03

Simplify Numerical Radical

We can simplify \(\sqrt[4]{81}\) to be 3, since 3^4 equals 81. The expression is now \(\frac{3x}{1}\).
04

Final Simplification

Given that any number divided by 1 remains the same number, we can simplify our expression to 3x.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. $$\text { a. } 3^{\frac{1}{2}} \square 3^{\frac{1}{3}}$$ $$\text { b. } \sqrt{7}+\sqrt{18} \square \sqrt{7}+18$$

Evaluate each expression without using a calculator. $$8^{\frac{2}{3}}$$

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$2 x^{4}\left(8 x^{4}+3 x\right)$$

Find all integers b so that the trinomial can be factored. $$ x^{2}+b x+15 $$
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