91Ó°ÊÓ

To start, let's understand what a fourth root is. The fourth root of a number is a value that, when multiplied by itself four times, gives the original number. For example, the fourth root of 16 is 2 because \((2 \cdot 2 \cdot 2 \cdot 2 = 16)\). In mathematical notation, the fourth root of a number x is written as \[ \sqrt[4]{x} \].
Fourth roots involve more complex calculations than square or cube roots, but the idea is the same. Splitting the problem into smaller parts often helps. For instance, if you have \[ \sqrt[4]{a \cdot b} \], you can split it into \(((\sqrt[4]{a}) \cdot (\sqrt[4]{b}))\). This property significantly simplifies calculations.

Exponent rules are essential for simplifying expressions. These rules help manage and combine powers of numbers and variables. Here are some key rules:

• Product Rule: \[ a^m \cdot a^n = a^{m+n} \]
• Quotient Rule: \[ \frac{a^m}{a^n} = a^{m-n} \]
• Power Rule: \[ (a^m)^n = a^{m \cdot n} \]
• Root Rule: \[ \sqrt[n]{a^m} = a^{m/n} \]

In our problem, the root rule is particularly useful. For example, \[ \sqrt[4]{3^5} \] can be simplified using the root rule \((3^{5/4})\). The same applies to variables: \[ \sqrt[4]{m^{19}} = m^{19/4} \] and \[ \sqrt[4]{n^{10}} = n^{5/2} \]. These simplifications help us express complex roots in a manageable form.

Prime factorization breaks down a number into its basic prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. For example, 243 can be broken down as follows:
243 is divisible by 3, so \[ \frac{243}{3} = 81 \].
81 is also divisible by 3, so \[ \frac{81}{3} = 27 \].
27 can be divided by 3, so \[ \frac{27}{3} = 9 \].
9 can be further divided by 3, so \[ \frac{9}{3} = 3 \].
Finally, 3 is a prime number. Putting it all together, we have \[ 243 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 3^5 \].
Understanding prime factorization is crucial for simplifying radical expressions. It allows us to express numbers like 243 in a format that makes it easier to apply rules of exponents and roots. For instance, we rewrite \[ \sqrt[4]{243} \] as \[ \sqrt[4]{3^5} \], simplifying the calculation to \[ 3^{5/4} \].