Problem 81 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ \sqrt{2}+\sqrt[3]{8} $$

Step-by-Step Solution

Answer

The sum of \( \sqrt{2} \) and \( \sqrt[3]{8} \) is \( \sqrt{2} + 2 \).

Step 1: Evaluate the square root

Evaluate the square root of 2. This cannot be simplified beyond \(\sqrt{2}\), so it will stay the same in the final answer.

Step 2: Evaluate the cube root

Evaluate the cube root of 8. The cube root of 8 is 2, because \(2^3 = 8\).

Step 3: Add the radicands

Now, add the results from the two previous steps. As you couldn't simplify \(\sqrt{2}\), and the cube root of 8 is 2, your answer is \( \sqrt{2} + 2\).

Key Concepts

Square Root

Understanding the square root is essential for simplifying radical expressions. A square root of a number 'x' is a value that, when multiplied by itself, gives the number 'x'. For example, the square root of 9 is 3, since 3 multiplied by 3 is 9. We denote the square root using the radical sign \( \sqrt{} \) followed by the number. In some cases, square roots can be simplified by factoring out perfect squares from inside the radical.

For instance, \( \sqrt{16} \) can be simplified because 16 is a perfect square, and thus \( \sqrt{16} = 4 \). However, when we encounter a square root of a prime number, such as \( \sqrt{2} \) in the original exercise, it cannot be simplified further because there are no square factors except 1.

Cube Root

The cube root is another type of radical expression, represented by \( \sqrt[3]{} \). The cube root of a number x, is a value that, when raised to the power of three (cubed), results in 'x'. For example, the cube root of 8 is 2, written as \( \sqrt[3]{8} = 2 \) because \(2^3 = 8\).

Understanding cube roots is crucial because not all roots are square roots, and recognizing the difference is key to solving radicals properly. Similar to square roots, if a number inside the cube root is a perfect cube, like 8, 27, or 64, the cube root can be evaluated to a whole number.

Radicals Addition

When simplifying expressions with radicals, we may sometimes have to perform operations like addition. However, radical addition follows specific rules. You can only add or subtract radicals that have the same index and radicand. It's similar to adding like terms in algebra.

For example \( \sqrt{2} + \sqrt{2} \) can be simplified to \( 2\sqrt{2} \) while \( \sqrt{2} + \sqrt{3} \) cannot be simplified because the radicands (numbers inside the root) are different. In the exercise given, \( \sqrt{2} \) and \( \sqrt[3]{8} \) are different types of roots and therefore cannot be added together directly; they are not 'like terms'.

Simplifying Square Roots

Simplifying square roots involves reducing the expression to its most basic form. To do so, you factor the number inside the square root and identify any perfect squares that can be taken out of the radical. Once any perfect squares are factored out, the square root is simplified.

For example, \( \sqrt{50} \) can be simplified to \( 5\sqrt{2} \) because 50 equals 25 times 2, and \( \sqrt{25} \) is 5. If there are no perfect square factors, as with \( \sqrt{2} \) in the exercise, the square root is already in its simplest form. Remember that the goal of simplifying square roots is to make the numbers smaller and the expression easier to understand and work with in further calculations.