Solve: Multiplication of Cube Roots with Nested Square Roots (√[3]{√3}·√[3]{√4})

Question

Complete the following exercise:

$\sqrt[3]{\sqrt{3}}\cdot\sqrt[3]{\sqrt{4}}=$

00:00 Solve the following problem

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

00:13 When there is a root of order (C) of root (B)

00:16 The result equals the root of the product of the orders

00:21 Apply this formula to our exercise

00:36 When we have a product of 2 numbers (A and B) in a root of order (C)

00:39 The result equals their product (A times B) in a root of order (C)

00:45 Apply this formula to our exercise

00:51 This is the solution

To solve $\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{4}}$ , we will convert these expressions into powers:

Step 1: Express each root as a power:
$\sqrt[3]{\sqrt{3}} = (\sqrt{3})^{1/3} = 3^{1/6}$
$\sqrt[3]{\sqrt{4}} = (\sqrt{4})^{1/3} = 4^{1/6}$

Step 2: Multiply the expressions using the property of exponents:
$3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6} = 12^{1/6}$

Therefore, the simplified expression is $\sqrt[6]{12}$ .

$\sqrt[6]{12}$