Evaluate. (root(4)(2))/(root(4)(162))=

Understanding the problem: First, let's understand the problem. We need to evaluate the expression which involves the fourth root of 2 divided by the fourth root of 162.Combining the roots: We can simplify the expression by combining the roots since they have the same index (fourth root). The property of roots we use here is root(n)(a)/root(n)(b) = root(n)(a/b).Applying the property: Now, let's apply the property to our expression: (root(4)(2))/(root(4)(162)) = root(4)(2/162).Simplifying the fraction: Next, we simplify the fraction 2/162 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 2/162 simplifies to 1/81.Rewriting the expression: Now we have root(4)(1/81). Since 81 is 3^4, we can rewrite the expression as root(4)(1/3^4).Using the property of roots: Using the property of roots that root(n)(a^n) = a, we can simplify root(4)(1/3^4) to 1/3.Final simplified value: Therefore, the final simplified value of the original expression is 1/3.

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Algebra 1

Solve advanced linear inequalities

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Q. Evaluate.

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\frac{\sqrt[4]{2}}{\sqrt[4]{162}}=

Understanding the problem: First, let's understand the problem. We need to evaluate the expression which involves the fourth root of $2$ divided by the fourth root of $162$ .
Combining the roots: We can simplify the expression by combining the roots since they have the same index (fourth root). The property of roots we use here is $\sqrt[4]{a}/\sqrt[4]{b} = \sqrt[4]{a/b}$ .
Applying the property: Now, let's apply the property to our expression: $\frac{\sqrt[4]{2}}{\sqrt[4]{162}} = \sqrt[4]{\frac{2}{162}}$ .
Simplifying the fraction: Next, we simplify the fraction $\frac{2}{162}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . So, $\frac{2}{162}$ simplifies to $\frac{1}{81}$ .
Rewriting the expression: Now we have $\sqrt[4]{\frac{1}{81}}$ . Since $81$ is $3^4$ , we can rewrite the expression as $\sqrt[4]{\frac{1}{3^4}}$ .
Using the property of roots: Using the property of roots that $\sqrt[4]{\frac{1}{3^4}} = \frac{1}{3}$ .
Final simplified value: Therefore, the final simplified value of the original expression is $\frac{1}{3}$ .