Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(2 \sqrt{27}-\sqrt{75}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(2 \sqrt{27}-\sqrt{75}\) is \(11\sqrt{3}\)

Step 1: Simplify the first square root

Find a perfect square that's a factor of 27. 9 is a perfect square and also a factor of 27. Hence, we write \(2\sqrt{27}\) as \(2\sqrt{9 \times 3}\). This further simplifies to \(2 \times \sqrt{9} \times \sqrt{3}\) which is \(6\sqrt{3}\).

Step 2: Simplify the second square root

Find a perfect square that's a factor of 75. 25 is a perfect square and also a factor of 75. Hence, we write \(-\sqrt{75}\) as \(-\sqrt{25 \times 3}\). This further simplifies to \(-\sqrt{25} \times \sqrt{3}\) which is \(-5\sqrt{3}\).

Step 3: Perform the subtraction

Now, subtract \(-5\sqrt{3}\) from \(6\sqrt{3}\). This equals to \(6\sqrt{3} - (-5\sqrt{3}) = 11\sqrt{3}\).

Key Concepts

Radical Expressions

Understanding radical expressions is essential in simplifying square roots. A radical expression is an expression that includes a square root, cube root, or higher roots. It’s important to recognize that the radical sign, \( \sqrt{} \), symbolizes the principal square root of a number. In the example provided, \(2 \sqrt{27}-\sqrt{75}\), we encounter two radical expressions that need to be simplified for further operations.

To approach the simplification, identifying factors of the number under the radical that are perfect squares can simplify the expression. Factoring allows us to 'break down' the radical into more manageable parts, making calculations easier and the expression clearer.

Perfect Squares

Perfect squares play a pivotal role in the simplification process of square roots. A perfect square is a number that can be expressed as the product of an integer with itself. Examples of perfect squares include 1, 4, 9, 16, 25, and so on. By recognizing that numbers like 27 and 75 have factors that are perfect squares—9 and 25, respectively—simplification becomes a matter of arithmetic.

When simplifying a square root, if you can identify a perfect square factor, you can take the square root of that factor out from under the radical sign. This is exactly what was done in the exercise, as the square root of 9, which is a perfect square, was taken out of the radical to be represented as 3 outside the radical.

Factoring in Algebra

Factoring is a foundational concept in algebra that involves breaking down numbers or expressions into a product of simpler numbers or expressions. It’s a key step in simplifying radical expressions and solving various algebraic equations. Factoring can simplify complex problems and reveal properties that are not readily apparent in the original expression.

For instance, in our exercise, 27 and 75 were factored to 9x3 and 25x3, respectively. By doing so, we utilized the property that the square root of a product can be expressed as the product of the square roots, which dramatically simplifies the computation steps. It is this factoring that reveals the perfect square factors and allows us to simplify the radical expressions effectively.

Arithmetic Operations with Radicals

When performing arithmetic with radicals, it's crucial to remember that only like radicals—radicals with the same index and radicand (the number under the root)—can be combined through addition or subtraction. Multiplication and division, on the other hand, can often be performed on different radicals and can help in simplifying expressions before combining like terms.

In our exercise, after simplifying the individual square roots, the subtraction operation is straightforward since both terms involve \(\sqrt{3}\). This allows for the coefficients, 6 and -5, to be combined just like with regular variables. The final result, \(11\sqrt{3}\), showcases the combination of arithmetic operations with radical terms and the power of simplification in streamlining the solution.