Simplify square root of square root of 243(x+1)

Solution:

Step 1:

Express $243(x+1)$ as $(3^2{)}^2\cdot 3(x+1)$.

Step 1.1:

Extract $81$ from $243$ to get $\sqrt{\sqrt{81\cdot 3(x+1)}}$.

Step 1.2:

Represent $81$ as $9^2$ to obtain $\sqrt{\sqrt{9^2\cdot 3(x+1)}}$.

Step 1.3:

Convert $9$ to $3^2$ resulting in $\sqrt{\sqrt{(3^2{)}^2\cdot 3(x+1)}}$.

Step 1.4:

Introduce additional parentheses to clarify the expression $\sqrt{\sqrt{((3^2{)}^2)\cdot (3(x+1))}}$.

Step 2:

Extract terms from under the radical sign to get $\sqrt{3^2\sqrt{3(x+1)}}$.

Step 3:

Calculate $3^2$ to simplify to $\sqrt{9\sqrt{3(x+1)}}$.

Step 4:

Rewrite $9\sqrt{3(x+1)}$ as $3^2\sqrt{3(x+1^2)}$.

Step 4.1:

Represent $9$ as $3^2$ to get $\sqrt{3^2\sqrt{3(x+1)}}$.

Step 4.2:

Express $1$ as $1^2$ to arrive at $\sqrt{3^2\sqrt{3(x+1^2)}}$.

Step 5:

Remove terms from under the radical to have $3\sqrt{\sqrt{3(x+1^2)}}$.

Step 6:

Recognize that any number to the power of one remains unchanged to simplify to $3\sqrt{\sqrt{3(x+1)}}$.

Step 7:

Rewrite the nested radical $\sqrt{\sqrt{3(x+1)}}$ as the fourth root to get $3\sqrt[4]{3(x+1)}$.

Knowledge Notes:

1. Simplifying nested radicals: When simplifying expressions like $\sqrt{\sqrt{a}}$, we can rewrite them as $\sqrt[4]{a}$ because the square root of a square root is equivalent to the fourth root.

2. Exponent rules: Remember that $(a^b{)}^c=a^{b\cdot c}$ and $a^b\cdot a^c=a^{b+c}$. These rules are used to simplify expressions with exponents.

3. Square roots and radicals: The square root of a product can be expressed as the product of the square roots, i.e., $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$.

4. Simplifying expressions: When simplifying expressions, it's often helpful to factor out perfect squares or cubes to make the radical simpler.

5. Rationalizing the denominator (not applicable in this problem, but a common technique in simplifying radicals): If a radical is in the denominator, we can multiply the numerator and the denominator by a suitable term to remove the radical from the denominator.