Problem 65 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. Your answer should not have any radicals in it. $$\sqrt{6}+\sqrt{2}(\sqrt{2}-\sqrt{3})$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt{6} + \sqrt{2}(\sqrt{2} - \sqrt{3})$$ without using a calculator. Answer: 2

Step 1: Distribute the square root of 2 within the parentheses

Multiply the square root of 2 by each term inside the parentheses. $$\sqrt{6} + \sqrt{2}(\sqrt{2} - \sqrt{3}) = \sqrt{6} + \sqrt{2} \cdot \sqrt{2} - \sqrt{2} \cdot \sqrt{3}$$

Step 2: Apply the rule of multiplication of radicals

To multiply the square roots, multiply the numbers inside the square roots and place the result inside a single square root. $$\sqrt{6}+ [\sqrt{2}\cdot\sqrt{2}] - [\sqrt{2}\cdot\sqrt{3}] = \sqrt{6} + \sqrt{2^2} - \sqrt{6}$$

Step 3: Simplify the square root of 2 squared

The square root of a number squared is equal to the number itself. $$\sqrt{6} + \sqrt{2^2} - \sqrt{6} = \sqrt{6} + 2 - \sqrt{6}$$

Step 4: Combine like terms

Subtract the two square root of 6 terms from each other. $$ \sqrt{6} + 2 - \sqrt{6} = 2 $$ So the simplified expression is $$2$$.

Key Concepts

Radical Expressions

A radical expression is an expression that includes a square root, cube root, or any other higher-order root. They're a way to represent that you're taking a root of a number or expression. For instance, the expression \(\sqrt{6} + \sqrt{2}(\sqrt{2} - \sqrt{3})\) contains radicals. Simplifying radical expressions often involves several steps, including distribution, combination of like terms, and applying operations like multiplication or division inside the root.

Simplification is the process of finding an equivalent expression that is simpler or more convenient to use. It may include removing the radical sign or combining terms to make the expression more compact. Students should keep in mind that just like how multiplication is the inverse operation of division, radicals relate to exponents as direct opposites. As such, the radical symbol signifies an operation that 'undoes' raising a number to a power.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{4}\) is 2 because \(2 \times 2 = 4\). In algebra, taking the square root is essential in solving equations and simplifying expressions. In the provided exercise, we see operations involving square roots such as \(\sqrt{2} \cdot \sqrt{2}\), which simplifies to 2 because the square root is the reverse operation of squaring.

Properties of Square Roots

Understanding properties is crucial when manipulating square roots. One property is that the square root of a square is the base number (\(\sqrt{x^2} = x\)). Also, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), meaning you can multiply within and take a single square root of the whole. However, it's important to note that \(\sqrt{a} + \sqrt{b}\) is not equal to \(\sqrt{a + b}\), an error often made by students.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions using a variety of techniques and rules. It's the bread and butter of solving algebra problems and includes distributing, factoring, combining like terms, and applying arithmetic operations consistent with algebraic rules.

In the provided example, distributing the \(\sqrt{2}\) across the parentheses is a part of algebraic manipulation. As seen from the exercise, using these techniques, the expression \(\sqrt{6} + \sqrt{2}(\sqrt{2} - \sqrt{3})\) is simplified to just 2. This process is helpful for tackling complex problems, and mastering algebraic manipulation can make the difference in understanding and solving algebraic expressions comprehensively. Engaging with algebraic manipulation frequently will build a foundation that greatly simplifies work with equations and expressions, especially when working with radicals and other more challenging concepts.