Problem 25 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{3} \cdot \sqrt{75} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt{3} \cdot \sqrt{75} \) is 15.

Step 1: Combine the square roots

By multiplication property of square roots, which states: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), the expression simplifies to \( \sqrt{3} \cdot \sqrt{75} = \sqrt{3 \cdot 75} \).

Step 2: Calculate the product under the square root

Calculate the product of 3 and 75, which is 225. So the expression now reads \( \sqrt{225} \).

Step 3: Simplify the square root

The square root of 225 is 15 because \(15^2 = 225\). Thus, the expression \( \sqrt{225} \) simplifies to 15. The expression \( \sqrt{3} \cdot \sqrt{75} \) is therefore equal to 15.

Key Concepts

Multiplication Property of Square Roots

When dealing with square roots, understanding the multiplication property of square roots is crucial. This property says that the product of two square roots is equal to the square root of the product of those numbers. In mathematical terms, if you have two numbers, say \( a \) and \( b \) , then this property is expressed as \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

This principle allows us to multiply roots before actually calculating the square root of each number, which often simplifies the solving process. For example, in the exercise \( \sqrt{3} \cdot \sqrt{75} \), instead of finding the square roots of 3 and 75 separately, we first combine them to find \( \sqrt{3 \cdot 75} = \sqrt{225} \). This makes the process of simplifying radical expressions more manageable and efficient.

Radical Expressions

A radical expression is an expression that includes a square root, cube root, or other higher roots. The square root, represented by the radical symbol \( \sqrt{\phantom{x}} \), is one of the most common radical expressions. Simplifying radical expressions like \( \sqrt{75} \) is often a matter of breaking the number down to its prime factors and identifying perfect squares within those factors.

Simplifying means to express the radical in the simplest form possible. For instance, \( \sqrt{75} \) can be simplified by recognizing that 75 equals 3 times 25, and since 25 is a perfect square, \( \sqrt{75} \) simplifies to \( 5 \sqrt{3} \). This step of recognizing perfect squares within the radicand—the number under the radical sign—makes simplification a lot easier.

Perfect Squares

Perfect squares play a pivotal role in simplifying square roots, as they are numbers that are the square of an integer, like \( 4 \) (which is \( 2^2 \) ), \( 9 \) ( \( 3^2 \) ), \( 16 \) ( \( 4^2 \) ), and so on. When a radicand is a perfect square, its square root will always be an integer.

In the context of simplification, by recognizing that a number is a perfect square, we can instantly find its square root without additional calculations. For example, during the simplification of \( \sqrt{225} \) we quickly discern that 225 is the square of 15, making the square root simply 15. Knowledge of perfect squares also helps in factoring radical expressions; we factor the radicand into a product that includes perfect squares whenever possible, making it effortless to simplify the root of that product.