Question

Simplify the radical expression. $$ \frac{1}{3} \sqrt{27} $$

Short answer

The simplified form of \( \frac{1}{3} \sqrt{27} \) is \( \sqrt{3} \).

Step-by-step solution

Perform Prime Factorization of the number under the root.

Carry out the prime factorization of the number under the root which is 27. The prime factorization of 27 is \(3 \cdot 3 \cdot 3\), which can be written as \(3^3\). The given expression hence becomes \( \frac{1}{3} \sqrt{3^3} \).

Simplify the Root

The rule of roots states that \(\sqrt{a^n} = a^{\frac{n}{2}}\) if \(n\) is an even number. Extending the rule, when \(n\) is odd (like 3 in this case), the result is \(a^{q}b^{\frac{1}{2}}\) where \(q\) is \(\frac{n-1}{2}\) and \(b\) is \(a\). Hence, \(\sqrt{3^3} = 3^{\frac{3-1}{2}} \cdot 3^{\frac{1}{2}} = 3 \cdot \sqrt{3}\). Substituting this in our expression, we get \( \frac{1}{3} \times 3 \cdot \sqrt{3} \).

Simplify the Expression

After the multiplication in the expression is performed in the previous step, \( \frac{1}{3} \times 3 \cdot \sqrt{3} \), the fraction \( \frac{1}{3} \times 3 \) equals to 1 and we are left with \( \sqrt{3} \).

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime numbers that, when multiplied together, give the original number. It is a crucial step in simplifying radical expressions because it reveals which factors can be taken out of the radical sign.

For example, to simplify the radical expression \( \frac{1}{3} \sqrt{27} \) from our exercise, we first need to perform prime factorization on the number 27. This process reveals that 27 is a product of 3's: \( 3 \cdot 3 \cdot 3 \) or \( 3^3 \) in exponential form. Understanding this basic concept allows us to identify the prime factors and sets the stage for further simplification under root rules.

Root Simplification

Root simplification involves applying specific rules to simplify the expressions under the radical sign. When the index of the root is 2, which makes it a square root, the rule \( \sqrt{a^n} = a^{\frac{n}{2}} \) is used if \(n\) is even. For odd numbers, like the 3 in our current example, you use the adjusted rule whereby \( \sqrt{a^n} = a^{q} \sqrt{a} \), with \( q \) being \( \frac{n-1}{2} \).

This method allows us to simplify \( \sqrt{3^3} \) into \( 3 \sqrt{3} \) by separating the root of a perfect square from the rest, making the radical expression easier to grasp and calculate.

Radical Expression Simplification

The simplification of radical expressions is the final step that brings together prime factorization and root simplification. It's about manipulating the expression to its simplest form.

Using our exercise as a case in point, once we've performed the prime factorization and root simplification, we address the multiplication of the fraction outside the radical with the number we extracted from under the radical sign. In our example, the \( \frac{1}{3} \) outside the radical cancels out with the 3 that was brought outside, leaving us with \( \sqrt{3} \) as the simplest form of the expression. This demonstrates that simplifying radicals can often lead to much cleaner and straightforward expressions, which are easier to work with in subsequent math problems and equations.