Question

Challenge Problems. Perform the indicated operation and simplify. $$\sqrt{80 a^{3}}-3 \sqrt{20 a^{3}}-2 \sqrt{45 a^{3}}$$

Short answer

-8\sqrt{5a^3}

Step-by-step solution

Factor inside the square roots

Begin by factoring each term inside the square roots to reveal perfect squares. Factor 80, 20, and 45 to get: \(\sqrt{80 a^{3}} = \sqrt{16 \cdot 5 \cdot a^{2} \cdot a}\), \(\sqrt{20 a^{3}} = \sqrt{4 \cdot 5 \cdot a^{2} \cdot a}\), and \(\sqrt{45 a^{3}} = \sqrt{9 \cdot 5 \cdot a^{2} \cdot a}\).

Simplify the square roots

Extract the square roots of the perfect squares from each term. For the first term: \(\sqrt{80 a^{3}} = \sqrt{16} \cdot \sqrt{5} \cdot \sqrt{a^{2}} \cdot \sqrt{a} = 4\sqrt{5}a \cdot \sqrt{a}\). Similarly, simplify the other terms to \(3 \sqrt{20 a^{3}} = 3 \cdot 2\sqrt{5}a \cdot \sqrt{a} = 6\sqrt{5}a \cdot \sqrt{a}\) and \(2 \sqrt{45 a^{3}} = 2 \cdot 3\sqrt{5}a \cdot \sqrt{a} = 6\sqrt{5}a \cdot \sqrt{a}\).

Combine like terms

Combine the like terms to simplify the expression. Since all terms are multiples of \(\sqrt{5}a \cdot \sqrt{a}\), they can be combined as \(4\sqrt{5}a\sqrt{a} - 6\sqrt{5}a\sqrt{a} - 6\sqrt{5}a\sqrt{a}\).

Perform the subtraction

Subtract the terms to get the simplified result: \(4\sqrt{5}a\sqrt{a} - 6\sqrt{5}a\sqrt{a} - 6\sqrt{5}a\sqrt{a} = (4 - 6 - 6)\sqrt{5}a\sqrt{a} = -8\sqrt{5}a\sqrt{a}\).

Write the final answer

Since \(\sqrt{a}\) cannot be further simplified, the final answer is \(-8\sqrt{5a^3}\).

Key concepts

Radical Expressions

Radical expressions are mathematical phrases that contain a radical symbol, such as the square root \( \sqrt{} \) or the cube root \( \sqrt[3]{} \) symbols. When simplifying radical expressions, the aim is to make the expression as simple as possible. This often involves identifying and extracting perfect squares, cubes, or higher-order 'perfect powers' from under the radical to simplify the expression.

For example, take the expression \( \sqrt{80 a^{3}} \). Here, 80 and \( a^{3} \) are under the radical. The goal is to find the largest perfect square that divides 80 and to express \( a^{3} \) as a product of squares whenever possible. In the original exercise, by factoring out perfect squares like 16 (from 80) and \( a^{2} \) (from \( a^{3} \)), we can simplify the original radical into simpler components.

Factorization

Factorization is the process of breaking down numbers or expressions into a product of simpler factors. It is a crucial concept in simplifying radical expressions and solving various types of algebraic problems. For example, factoring the number 80 into \( 16 \cdot 5 \) reveals that 16 is a perfect square and can be easily taken out of the square root. This process is applied to both numerical and variable components within an expression.

In our given problem, we factored out perfect squares from under the radicals to simplify each term before combining like terms. Factorization not only helps in such simplification but also in identifying common factors which can lead to further reduction of the expression. It is an essential skill to master when working with radical expressions.

Combining Like Terms

Combining like terms is a fundamental method in algebra, used to simplify expressions. Like terms are terms that have the exact same variable parts, raised to the same power. It allows them to be added or subtracted from each other. When simplifying radical expressions, after the factorization process, it is common to encounter like radical terms.

In the given exercise, each term after simplification contained the factor \( \sqrt{5}a \sqrt{a} \), which meant they could be combined. This process involves adding or subtracting coefficients of like terms, which in our example led to \( (4 - 6 - 6)\sqrt{5}a\sqrt{a} \), simplifying further to \( -8\sqrt{5a^3} \). Thus, knowing how to identify and combine like terms is key to simplifying complex algebraic and radical expressions effectively.