Simplify square root of square root of 243(x+1)
The given problem is asking you to perform two consecutive square root operations on the expression $243(x+1)$. You are first supposed to simplify by finding the square root of the quantity $243(x+1)$and then subsequently find the square root of the resulting expression. The entire operation should be simplified to its most basic form if possible.
$\sqrt{\sqrt{243 \left(\right. x + 1 \left.\right)}}$
Answer
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:
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.
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.
Square roots and radicals: The square root of a product can be expressed as the product of the square roots, i.e., $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$.
Simplifying expressions: When simplifying expressions, it's often helpful to factor out perfect squares or cubes to make the radical simpler.
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.
