Problem 112 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \sqrt{288 t}+\sqrt{80 t}-\sqrt{128 t} $$

Step-by-Step Solution

Answer

The simplified expression is \(4\sqrt{2t} + 4\sqrt{5t}\).

Step 1: Simplify Each Radical Expression

First, we simplify each radical expression by factoring out the largest perfect square from under the square root. For \(\sqrt{288t}\), factor 288 as \(144 \times 2\). Thus, \(\sqrt{288t} = \sqrt{144 \times 2t} = \sqrt{144} \times \sqrt{2t} = 12\sqrt{2t}\).For \(\sqrt{80t}\), factor 80 as \(16 \times 5\). Thus, \(\sqrt{80t} = \sqrt{16 \times 5t} = \sqrt{16} \times \sqrt{5t} = 4\sqrt{5t}\).For \(\sqrt{128t}\), factor 128 as \(64 \times 2\). Thus, \(\sqrt{128t} = \sqrt{64 \times 2t} = \sqrt{64} \times \sqrt{2t} = 8\sqrt{2t}\).

Step 2: Combine Like Radical Terms

Now that each radical has been simplified, we look for like terms.Combine \(12\sqrt{2t}\) and \(8\sqrt{2t}\) since they have the same radical part:\(12\sqrt{2t} - 8\sqrt{2t} = (12 - 8)\sqrt{2t} = 4\sqrt{2t}\).Note that \(4\sqrt{5t}\) does not combine with this term because it has a different radical part.

Step 3: Write the Final Simplified Expression

After combining like terms, the expression is:\[4\sqrt{2t} + 4\sqrt{5t}\]This is already simplified as no further combination or simplification can be done.

Key Concepts

Addition of Radicals

Adding radicals is a bit like adding variables in algebra. To add radical expressions, the radicals must have the same index and radicand. In simpler terms, the parts under the square root sign must be identical. For instance, to add \(a\sqrt{b}\) and \(c\sqrt{b}\), the expressions \(a\) and \(c\) can be combined, resulting in \((a+c)\sqrt{b}\). This is similar to how you would add \(3x\) and \(2x\) to get \(5x\).
However, if the radical parts are different, they cannot be directly combined, just as \(3x\) can't be combined with \(2y\). In the provided exercise, we simplify the radicals first to see if they can be combined. After simplification, if like radicals exist, we sum up their coefficients. This rule helps us keep complex expressions tidy and accurate.

Subtraction of Radicals

Subtraction of radicals follows the same principles as addition. First, ensure the radicals are in their simplest form by simplifying them. Once simplified, you check if any of the radicals can be grouped together. Only radicals with the same index and radicand can be subtracted from each other.
In the exercise example, once simplified, \(12\sqrt{2t}\) and \(8\sqrt{2t}\) were found to have matching radical parts. So we could subtract their coefficients like this: \(12 - 8\), resulting in \(4\sqrt{2t}\). It's just like subtracting \(12x\) by \(8x\), which gives \(4x\). Remember, you cannot subtract radicals with different radicands, much like you wouldn’t subtract \(4x\) and \(3y\).

Simplifying Radicals

Simplifying radicals is a foundational process in working with radical expressions. The main goal is to express the radical in its simplest form, which often involves factoring out perfect squares. A perfect square is a number like 4, 9, 16, etc., whose square root is a whole number.
For example, simplifying \(\sqrt{288t}\) involves factoring 288 into \(144 \times 2\), where 144 is a perfect square. This lets us write \(\sqrt{288t} = \sqrt{144} \times \sqrt{2t} = 12\sqrt{2t}\). This simplification is crucial because it makes addition or subtraction possible by matching like terms. The process of simplifying helps in uncovering the underlying structures of an expression, setting the stage for further operations.

Rationalizing the Denominator

Even though the exercise does not directly involve rationalizing a denominator, it's important to understand it as it often comes up when radicals are involved. This process is used to eliminate radicals from the denominator of a fraction.
To rationalize a radical in the denominator, multiply both the numerator and the denominator by a conjugate or another radical to create a perfect square in the denominator. For square roots, this often means multiplying by the same root that appears in the denominator. For example, if you have \(\frac{1}{\sqrt{2}}\), multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\) to get \(\frac{\sqrt{2}}{2}\).
Rationalizing helps make expressions simpler and more acceptable in final answers, as it's often considered standard form to have no radicals in a denominator.