Simplify the Radical Expression 10 root of 3^5
The question asks you to perform a simplification on a radical expression. The expression given is the tenth root (indicated by "root of") of 3 raised to the 5th power (3^5). Simplifying this expression would require you to apply the properties of exponents and radicals to express the radical in its simplest form. It may also involve breaking down the exponent under the radical to see if any factors can be extracted from under the radical sign, based on the index of the root (which is 10 in this case).
$\sqrt[10]{3^{5}}$
Answer
Solution:
Step 1:
Compute $3^5$ to find the value under the radical. $\sqrt[10]{243}$
Step 2:
Express $243$ as a power of $3$. $\sqrt[10]{3^5}$
Step 3:
Transform $\sqrt[10]{3^5}$ into a nested radical. $\sqrt{\sqrt[5]{3^5}}$
Step 4:
Simplify the nested radical by extracting terms. $\sqrt{3}$
Step 5:
Present the simplified expression in its various forms.
Exact Form: $\sqrt{3}$ Decimal Form: $1.73205080\ldots$
Knowledge Notes:
The problem involves simplifying a radical expression, which is a mathematical expression containing a radical symbol (鈭? with an index. In this case, the index is 10, and we are dealing with a 10th root. The steps to simplify the expression are as follows:
Exponentiation: The first step is to raise the base number (3) to the exponent (5). In LaTeX, this is written as $3^5$.
Radical Notation: The next step is to express the number 243 as a power of 3, which is its prime factorization. In LaTeX, this is written as $\sqrt[10]{3^5}$.
Nested Radicals: The expression $\sqrt[10]{3^5}$ can be rewritten as a nested radical, which is a radical within another radical. This is done by recognizing that the 10th root of something can be written as the square root of the 5th root of that same thing. In LaTeX, this is written as $\sqrt{\sqrt[5]{3^5}}$.
Simplification of Radicals: The nested radical can be simplified by extracting terms from under the radical sign. Since the 5th root of $3^5$ is just 3, the expression simplifies to the square root of 3. In LaTeX, this is written as $\sqrt{3}$.
Expression in Different Forms: The final simplified expression can be presented in its exact form, which is $\sqrt{3}$, or in decimal form, which is an approximation of the square root of 3. The decimal form can be written out to several decimal places, as shown.
The relevant knowledge for solving this problem includes understanding the properties of exponents, how to simplify radical expressions, and the concept of nested radicals. Additionally, one must be familiar with the different ways to present mathematical expressions, such as exact form and decimal form.
