# Root of a root - Examples, Exercises and Solutions

## Square Roots

When we encounter an exercise in which there is a root applied to another root, we will multiply the order of the first root by the order of the second and the order obtained (the product of both) will be raised as a root in our number (as generally power of a power)
Let's put it this way:

## Practice Root of a root

### Exercise #1

Solve the following exercise:

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

### Step-by-Step Solution

To simplify the given expression, we will use two laws of exponents:

A. Definition of the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

B. Law of exponents for an exponent on an exponent:

$(a^m)^n=a^{m\cdot n}$

Let's begin simplifying the given expression:

$\sqrt[4]{\sqrt[3]{3}}= \\$We will use the law of exponents shown in A and first convert the roots in the expression to exponents, we will do this in two steps - in the first step we will convert the inner root in the expression and in the next step we will convert the outer root:

$\sqrt[4]{\sqrt[3]{3}}= \\ \sqrt[4]{3^{\frac{1}{3}}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}=$We continue and use the law of exponents shown in B for performing the exponent on the term (in parentheses) which is in the exponent, then we will perform the multiplication of the exponents which are in the exponent term:

$(3^{\frac{1}{3}})^{\frac{1}{4}}= \\ 3^{\frac{1}{3}\cdot\frac{1}{4}}=\\ 3^{\frac{1\cdot1}{3\cdot4}}=\\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}}$ In the final step we return to writing the root, that is - back, using the law of exponents shown in A (in the opposite direction),

Let's summarize the simplification of the given expression:

$\sqrt[4]{\sqrt[3]{3}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}= \\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}}$Therefore, note that the correct answer (most) is answer D.

### Exercise #2

Solve the following exercise:

$\sqrt{\sqrt{2}}=$

### Video Solution

$\sqrt[4]{2}$

### Exercise #3

Solve the following exercise:

$\sqrt[5]{\sqrt[3]{5}}=$

### Video Solution

$\sqrt[15]{5}$

### Exercise #4

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

### Exercise #5

Solve the following exercise:

$\sqrt[6]{\sqrt{2}}=$

### Video Solution

$\sqrt[12]{2}$

### Exercise #1

Solve the following exercise:

$\sqrt[]{\sqrt{8}}=$

### Video Solution

$8^{\frac{1}{4}}$

### Exercise #2

Solve the following exercise:

$\sqrt[6]{\sqrt{2}}=$

### Video Solution

$\sqrt[12]{2}$

### Exercise #3

Solve the following exercise:

$\sqrt{\sqrt{12}}=$

### Video Solution

$\sqrt[4]{12}$

### Exercise #4

Solve the following exercise:

$\sqrt{\sqrt{4}}=$

### Video Solution

$\sqrt{2}$

### Exercise #5

Solve the following exercise:

$\sqrt[7]{\sqrt{2}}=$

### Video Solution

$\sqrt[14]{2}$

### Exercise #1

Solve the following exercise:

$\sqrt[3]{\sqrt{729}}=$

3

### Exercise #2

Solve the following exercise:

$\sqrt[3]{\sqrt[3]{512}}=$

2

### Exercise #3

Solve the following exercise:

$\sqrt[5]{\sqrt[]{1024}}=$

2

### Exercise #4

Solve the following exercise:

$\sqrt{\sqrt{625}}=$

5

### Exercise #5

Solve the following exercise:

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