Problem 10 - Elementary Algebra

Question

Write the principal and secondary square roots of each number. $$ 100 $$

Step-by-Step Solution

Answer

Answer: The principal square root of 100 is 10, and the secondary square root of 100 is -10.

Step 1: Find the principal square root of 100

To find the principal square root of 100, we need to find the positive number that when multiplied by itself, gives us 100. In this case, we have: $$ \sqrt{100} = 10 $$ Because \(10 \times 10 = 100\).

Step 2: Find the secondary square root of 100

To find the secondary square root of 100, we need to find the negative number that when multiplied by itself, gives us 100. In this case, we have: $$ -\sqrt{100} = -10 $$ Because \((-10) \times (-10) = 100\).

Step 3: Write the final answer

Both the principal and secondary square roots of 100 are found: $$ \sqrt{100} = 10 $$ $$ -\sqrt{100} = -10 $$

Key Concepts

Principal Square Root

Understanding the principal square root is essential when working with square roots. It refers to the non-negative root of a given number. When you see the symbol \( \sqrt{x} \), it is asking for the principal square root of \( x \). For the number 100, the principal square root is calculated by seeking a positive number that, when multiplied by itself, returns the original number. Thus, \( \sqrt{100} = 10 \), since \( 10 \times 10 = 100 \).

This concept is vital because it forms the basis of understanding other types of roots and will be used frequently in various mathematical operations. Remember, the principal square root will always be the positive solution.

Secondary Square Root

Opposite to the principal square root is the secondary square root, which represents the negative root of a number. While the principal square root is represented by \( \sqrt{x} \), the secondary square root is denoted as \( -\sqrt{x} \). For the number 100, \( -\sqrt{100} = -10 \) because when you multiply \( -10 \) by itself, \( (-10) \times (-10) = 100 \).

Students should note that every positive number has two square roots – the principal (positive) and the secondary (negative) – and both are equally important in the realm of complex calculations.

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 100 is a perfect square because it equals \( 10 \times 10 \). Knowing if a number is a perfect square is helpful when performing square roots because the square root of a perfect square is always an integer. This is especially important in simplifying roots.

Other examples of perfect squares include \( 1 = 1 \times 1 \), \( 4 = 2 \times 2 \), \( 9 = 3 \times 3 \), and so on. A quick way to recognize some perfect squares is by remembering that they end in either 0, 1, 4, 5, 6, or 9.

Square Root Operations

When performing operations with square roots, like addition, subtraction, multiplication, and division, there are specific rules to follow. Just like simple arithmetics, roots can only be added or subtracted when they have the same radicand (the number under the root sign). For example, \( \sqrt{9} + \sqrt{9} = 2\sqrt{9} \) which simplifies to \( 2 \times 3 \), because \( \sqrt{9} \) is 3.

For multiplication and division, you can operate across different radicands, such as \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \) and \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). It's essential to understand these operations as they form the basis for more complex algebraic expressions involving roots.