Problem 81 - Algebra 1

Question

Simplify the radical expression. $$\frac{1}{2} \sqrt{52}$$

Step-by-Step Solution

Answer

The radical expression \(\frac{1}{2} \sqrt{52}\) simplifies to \(\sqrt{13}\).

Step 1: Isolate the radical

Begin by isolating the radical expression. This means focussing on the part under the square root, which in this case, is \(52\).

Step 2: Break down the number into factors

Break down \(52\) into its factors. Fifty-two can be expressed as \( 4 \times 13\), where \(4\) is the perfect square.

Step 3: Apply the square root

Apply the square root to the perfect square, which is 4. The square root of \(4\) equals \(2\). So, \(\sqrt{4 \times 13}\) becomes \(2 \sqrt{13}\).

Step 4: Simplify the expression

Finally, incorporate the fraction \(\frac{1}{2}\) back into the expression. This gives \(\frac{1}{2} \times 2 \sqrt{13}\), which equals \(\sqrt{13}\).

Key Concepts

Isolate the Radical

When simplifying radical expressions, the first step is often to 'isolate the radical'. This involves focusing solely on the expression under the radical sign, separating it from other parts of the equation or expression. For example, in the expression \(\frac{1}{2} \sqrt{52}\), the radical is \(\sqrt{52}\). By isolating it, we can concentrate on simplifying just that section without the distraction of the coefficient \(\frac{1}{2}\) outside the radical.

Factorization

Factorization plays a pivotal role in simplifying radicals. It involves breaking down a number into its prime factors, or in some cases, identifying a factor that is a perfect square. This is particularly useful because the square root of a perfect square is always an integer. As seen in our example, the number \(52\) factors into \(4 \times 13\), where \(4\) is the perfect square. Recognizing these factors enables further simplification of the radical expression.

Square Roots

The square root is a fundamental operation in mathematics, indicating what number, when multiplied by itself, will yield the original number. Simplifying square roots involves finding such an integer if one exists. In our example, the square root of the perfect square \(4\), a factor of \(52\), is \(2\). Understanding how to extract square roots of perfect squares is essential for radical simplification.

Radical Simplification

Radical simplification refers to the process of simplifying expressions containing a radical sign to their simplest form. The goal is to ensure that the radicand, the number under the radical sign, doesn't contain any perfect square factors. After factorization and taking square roots of any perfect squares, the simplified radical contains only prime factors or an integer multiplied by a radical that can no longer be simplified. In the given exercise, after isolating the radical and factorizing \(52\), applying the square root to the perfect square, and multiplying by the outside fraction, the expression simplifies to \(\sqrt{13}\), which is in its simplest radical form.