Problem 120 - 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. $$ (2 \sqrt{3 t}+5)^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(12t + 20\sqrt{3t} + 25\).

Step 1: Understand the Expression

The given expression is the square of a binomial: \((2\sqrt{3t} + 5)^2\). This means we will apply the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Identify \(a = 2\sqrt{3t}\) and \(b = 5\).

Step 2: Square the First Term

First, compute \(a^2\), which is \((2\sqrt{3t})^2\). Squaring this term: \([2^2 \times (\sqrt{3t})^2] = 4 \times 3t = 12t\).

Step 3: Multiply the Two Terms by 2

Next, compute \(2ab\), which is \(2 \times (2\sqrt{3t}) \times 5\). Calculate: \(2 \times 2 \times 5 \times \sqrt{3t} = 20\sqrt{3t}\).

Step 4: Square the Last Term

Finally, compute \(b^2\), which is \(5^2\). This simplifies to: \(5^2 = 25\).

Step 5: Combine All Terms

Combine all the computed terms: \(a^2 + 2ab + b^2 = 12t + 20\sqrt{3t} + 25\). This is the simplified form of the given expression.

Key Concepts

Binomial Expansion

In algebra, a binomial expansion is used to expand expressions that are in the form \((a+b)^n\). Here, "binomial" refers to an algebraic expression with two terms, a and b, raised to a power n. To expand binomials, you can use the Binomial Theorem, which gives a quick way to expand powers of binomials, but for small powers, hand calculation methods can also be easy.

For instance, if n is 2, as in \((a + b)^2\), we can apply a simple formula: \[ (a + b)^2 = a^2 + 2ab + b^2 \]

This is derived directly from multiplying the binomial by itself.

• The term \(a^2\) represents the square of the first term.
• \(2ab\) represents twice the product of the two terms.
• \(b^2\) is the square of the second term.

By breaking it down into these parts, it becomes easy to manage and solve each component.Using these principles significantly simplifies calculations in algebra.

Squaring Binomials

Squaring a binomial means multiplying it by itself. When you are faced with a problem such as \((2\sqrt{3t} + 5)^2\), you are essentially applying the binomial expansion formula for when \(n=2\).

The process involves three main steps:

• **Square the first term**: Here, the first term is \(2\sqrt{3t}\). When you square it, you do \(2\sqrt{3t}\)^2 = 4 \(\times\) (3t) = 12t.
• **Multiply the terms and double it**: The second part of the formula, \(2ab\), means you take the product of the two terms (\(2\sqrt{3t}\) and \(5\)), and multiply it by 2, resulting in \(20\sqrt{3t}\).
• **Square the second term**: Finally, you square \(b\), which is 5. Squaring it gives you 25.

After you carry out these operations, you combine everything: \(12t + 20\sqrt{3t} + 25\). This gives you the expanded result of the original expression. The beauty of this approach is in its simplicity once you identify a, b, and follow the rules.

Radical Expressions

Radical expressions involve roots, like square roots or cube roots. A radical sign (√) denotes these types of expressions. In the expression \(2\sqrt{3t}\), \(\sqrt{3t}\) is a radical expression. Understanding how to handle these is crucial for expanding on them algebraically.

Radical expressions often need simplification or evaluation as a part of larger expressions. Here's how you can approach them:

• **Simplifying radials**: Try to simplify the expression under the radical, if possible. For example, if \(\sqrt{12t}\) were involved, you could simplify it to \(2\sqrt{3t}\).
• **Rationalizing**: If a radical is in the denominator of a fraction, you often need to rationalize it. This means restructuring the expression so that there are no radicals in the denominator.
• **Combining with other terms**: As seen in binomial expansion, sometimes radical expressions are part of a larger problem requiring you to multiply or combine them with other terms.

Mastering radical expressions is essential for simplifying algebraic expressions and solving equations in an efficient manner.