Solve the Expression: Cube Root of Square Root of 64 Times Square Root of 64

Solve the following exercise:

$\sqrt[3]{\sqrt{64}}\cdot\sqrt{64}=$

Video Solution

Solution Steps

00:00 Solve the following problem

00:03 A 'regular' root is of the order 2

00:10 When we have a root of the order (B) in a root of the order (A)

00:14 We obtain a root of the order of the product of orders (B times A)

00:20 Let's apply this formula to our exercise

00:28 Calculate the product of orders

00:36 Break down 64 to 2 to the power of 6

00:40 Break down 64 to 8 squared

00:43 A root of the order 6 cancels out the power of 6

00:46 The root cancels out the square

00:50 This is the solution

Step-by-Step Solution

To solve the expression $\sqrt[3]{\sqrt{64}}\cdot\sqrt{64}$ , we follow these steps:

Step 1: Express $\sqrt{64}$ as a power:
Since $\sqrt{64} = 64^{1/2}$ and $64 = 2^6$ , substituting gives $(2^6)^{1/2} = 2^{6 \cdot 1/2} = 2^3 = 8$ .
Step 2: Express $\sqrt[3]{\sqrt{64}}$ as a power:
Since from Step 1, $\sqrt{64} = 2^3 = 8$ , then $\sqrt[3]{\sqrt{64}} = \sqrt[3]{8}$ .
Now, $\sqrt[3]{8} = 8^{1/3}$ and $8 = 2^3$ , so $(2^3)^{1/3} = 2^{3 \cdot 1/3} = 2^1 = 2$ .
Step 3: Multiply the simplified expressions:
We now have $\sqrt[3]{\sqrt{64}} = 2$ and $\sqrt{64} = 8$ .
Thus, $\sqrt[3]{\sqrt{64}} \cdot \sqrt{64} = 2 \cdot 8 = 16$ .

Therefore, the solution to the problem is $16$ .