Finding the Greatest Value: Mathematical Comparison Exercise

Video Solution

Solution Steps

00:00 Choose the largest value

00:03 A 'regular' root raised to the second power

00:07 Combine into one root by multiplying the orders together

00:11 This is the value of the first expression

00:19 Apply the same method to this expression

00:24 They are all equal, and this is the solution

Step-by-Step Solution

To determine if one of these values is the largest or if they are equal, we will express each expression as a power of 5:

First, consider $\sqrt[6]{\sqrt{5}}$ . This is equivalent to $(\sqrt{5})^{1/6} = (5^{1/2})^{1/6}$ . Applying the power rule, this is $5^{1/12}$ .
Next, consider $\sqrt[12]{5}$ , which is already expressed as a power: $5^{1/12}$ .
Finally, consider $\sqrt[4]{\sqrt[3]{5}}$ . This can be rewritten as $(\sqrt[3]{5})^{1/4} = (5^{1/3})^{1/4}$ . Again, using the power rule, this is $5^{1/12}$ .

All three expressions simplify to $5^{1/12}$ . Therefore, all values are equal. The correct choice is:

All values are equal.