Problem 67 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ \sqrt{18 t}+\sqrt{300 t}-\sqrt{243 t} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 3\sqrt{2t} + \sqrt{3t} \).

Step 1: Factor Inside the Radicals

Start by factoring the numbers inside the radicals to identify any perfect squares. \( \sqrt{18t} = \sqrt{9 \times 2 \times t} = \sqrt{9} \times \sqrt{2t} = 3\sqrt{2t} \) \( \sqrt{300t} = \sqrt{100 \times 3 \times t} = \sqrt{100} \times \sqrt{3t} = 10\sqrt{3t} \)\( \sqrt{243t} = \sqrt{81 \times 3 \times t} = \sqrt{81} \times \sqrt{3t} = 9\sqrt{3t} \)

Step 2: Combine Like Radicals

Look for like radicals that can be combined. The like radicals are \(3\sqrt{2t}\), \(10\sqrt{3t}\), and \(-9\sqrt{3t}\). We rewrite and simplify as follows: \( 3\sqrt{2t} + 10\sqrt{3t} - 9\sqrt{3t} \) Combine \(10\sqrt{3t}\) and \(-9\sqrt{3t}\) because they are like terms: \( 3\sqrt{2t} + (10 - 9)\sqrt{3t} = 3\sqrt{2t} + 1\sqrt{3t} \) The expression reduces to \( 3\sqrt{2t} + \sqrt{3t} \).

Step 3: Final Simplified Expression

Since the radicals \(\sqrt{2t}\) and \(\sqrt{3t}\) are not like terms, the expression is already in its simplest form. The final simplified expression is \( 3\sqrt{2t} + \sqrt{3t} \).

Key Concepts

Combining Like Terms

Combining like terms isn't just for algebraic expressions; it applies to radicals too! When simplifying radicals, finding like terms means identifying terms where both the radicand (the number under the square root) and the variable parts are identical.
For example, consider the radicals in the expression \( \sqrt{3t} \). Both \(10\sqrt{3t}\) and \(-9\sqrt{3t}\) have the same radicand \(t\), allowing them to be combined. When you add these, you add their coefficients—just like in algebraic expressions.

• Find terms with the same radicand.
• Make sure variable parts match as well.
• Add or subtract the coefficients, keeping the radical part unchanged.

Combining these terms can drastically simplify your expression.

Factoring Inside the Radicals

Factoring inside radicals is a crucial step for simplification. It involves breaking down the number inside the square root into its prime factors. This process helps identify any perfect squares, which can be moved outside the radical.
Consider the radical \( \sqrt{18t} \). First, factor \(18\) as \(9 \times 2\) because 9 is a perfect square and can be expressed as \(3 \times 3\). The expression becomes \( \sqrt{9} \times \sqrt{2t} = 3\sqrt{2t} \).
Repeat this process for other terms:

• Identify any perfect squares.
• Factor the expression fully.
• Extract the square root of perfect squares.

This simplifies the terms and makes it easier to identify and combine like radicals later.

Radical Expressions

Radical expressions involve roots, often square roots, but can also include cube roots, fourth roots, etc. When working with these expressions, the goal often is to simplify them by reducing the underlying numbers and variables.
Simple radical expressions, like \( \sqrt{x} \), are straightforward, but when combined with coefficients or added to each other, it can become more complex. In our example, radical expressions are filtered down to like terms or simplified so they are easier to manage.
Always remember to:

• Look for simplifications.
• Identify and combine like terms when possible.
• Understand that not all radicals can be combined—only those with the exact matching radicands and variables.

This helps in honing your skills for manipulating and simplifying complex radical expressions.