Problem 34 - Algebra 1

Question

Simplify the expression. $$\frac{\sqrt{32}}{\sqrt{25}}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{\sqrt{32}}{\sqrt{25}}\) is \(\frac{4\sqrt{2}}{5}\).

Step 1: Simplify the Square Roots

Evaluate \(\sqrt{32}\) and \(\sqrt{25}\) separately, to simplify the expression. The square root of 32 is \(4\sqrt{2}\) because \(32 = 16 * 2\) and the square root of 16 is 4. The square root of 25 is 5 because \(5 * 5 = 25\). So the expression simplifies to: \(\frac{4\sqrt{2}}{5}\)

Step 2: Express Answer in Simplest Form

\(\frac{4\sqrt{2}}{5}\) is in simplest form, as \(4\sqrt{2}\) cannot be further reduced and 5 is a prime number.

Key Concepts

Radical Expressions

Understanding radical expressions is essential for students mastering algebra. A radical expression is an expression that includes a square root, cube root, or any higher-order root. The symbol for the square root is \( \sqrt{ } \), and it's used to indicate that you're looking for a number which, when multiplied by itself, gives you the value inside the radical.

For example, consider the expression \( \sqrt{32} \). To simplify this, one must look for 'perfect squares'—numbers that are squares of integers—within the radicand, which is the number under the square root symbol. In this case, 32 can be broken down into 16 and 2, with 16 being a perfect square. This gives \( \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \). Similarly, \( \sqrt{25} \) simplifies to 5, as 25 is a perfect square of 5. Recognizing perfect squares like 4, 9, 16, 25, and so on, will make simplifying square roots much easier.

Simplify Expressions

One of the fundamental skills in algebra is learning how to simplify expressions. This means reducing an expression to its most basic form without changing its value. The simplification process often involves a variety of steps, including combining like terms, factoring, and canceling common factors.

When dealing with square roots, the goal is to express the square root as much as possible in terms of square numbers, and then simplify. For the expression \( \frac{\sqrt{32}}{\sqrt{25}} \), the simplification process involves breaking down the radicand, 32, into a product of 16 and 2, because 16 is a square number. The process leads to a simpler form, \( \frac{4\sqrt{2}}{5} \), where \( 4\sqrt{2} \) cannot be reduced further, and the denominator, 5, is already in its simplest prime form. Simplification doesn't alter the value of the expression; it merely presents it in a clearer and often more usable form.

Arithmetic Operations with Radicals

Working with arithmetic operations on radical expressions involves the same fundamental principles as other arithmetic operations, but with an additional layer of complexity due to the presence of roots. Common operations include addition, subtraction, multiplication, and division, as seen when simplifying radical expressions.

In the given example, we are performing a division of two square roots. When dividing radicals, especially square roots, with the same index (which in the case of square roots is 2), the process is akin to dividing the numbers directly if they are already in simplest radical form. The result, \( \frac{4\sqrt{2}}{5} \), demonstrates this operation, where you can regard the numerators and denominators as separate entities and simplify them as such, always keeping in mind to look for and simplify perfect square factors within radicands.