Problem 35 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Check the results by squaring each root. $$ \sqrt{121} $$

Step-by-Step Solution

Answer

The square root of 121 is \( \pm 11 \).

Step 1: Finding the square root

First, find the square root of 121. The square root of 121 is \( \pm 11\), which means that both -11 and +11 are the roots of 121.

Step 2: Verify the roots

Next, verify these roots by squaring each one. The square of -11 and +11 are both 121, which confirms that these are indeed the roots of 121.

Key Concepts

Simplifying Square Roots

Understanding how to simplify square roots is foundational in algebra and higher mathematics. Simply put, the square root of a number answers the question, 'What number, when multiplied by itself, gives me the original number?' When simplifying square roots, the goal is to find the root that is a whole number whenever possible. This is often straightforward for perfect squares, like 121, which are the product of a whole number multiplied by itself.

For numbers that are not perfect squares, simplification involves factoring out perfect squares from the radicand (the number inside the square root) and separating them from the non-perfect squares. For example, \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\). Here, we've identified that 25 is a perfect square that can be taken out of the square root, leaving \(\sqrt{2}\) which cannot be simplified further. This technique reduces the radicand to its simplest form, making the expression easier to work with in equations.

Perfect Squares

A perfect square is a number that has an integer square root. In the context of simplifying square roots, it's imperative to be able to recognize perfect squares since they can be directly transformed into their root integers. Examples of perfect squares include 1 (\(\sqrt{1}=1\)), 4 (\(\sqrt{4}=2\)), 9 (\(\sqrt{9}=3\)), up to numbers like 121 (\(\sqrt{121}=11\)), and beyond.

Knowing the list of perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, etc.) helps in factoring numbers under a square root sign. Factoring a radicand into perfect square factors can vastly simplify square root expressions, turning a complex calculation into a more manageable one. When no perfect square factor exists, the square root expression is already in its simplest form.

Square Root Properties

The properties of square roots are rules that help us handle square roots in various mathematical scenarios. Key properties include:

• The Product Property of Square Roots: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). This means you can multiply two square roots and then find the square root of the product, or find two square roots separately and then multiply them.
• The Quotient Property of Square Roots: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) when \(b eq 0\). This property says you can take the square root of a fraction by taking the square root of the numerator and the square root of the denominator separately.
• Even and Odd Powers: Only even powers of variables can be taken out of the square root sign as a whole number, whereas odd powers will have one less power outside the root, while one remains inside. For example, \(\sqrt{x^4} = x^2\) and \(\sqrt{x^3} = x\sqrt{x}\).
• The Square of a Square Root: \((\sqrt{a})^2 = a\), which means that when you square a square root, you get the original number inside the root back.

These properties not only guide us in simplifying square roots but also in performing operations involving square roots when solving equations, enabling us to manipulate and solve for variables under the root.