Problem 22 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ z^{2}=169 $$

Step-by-Step Solution

Answer

z = ±13.

Step 1 - Understand the Square Root Property

The square root property states that if a^2 = b, then a = ±√b. This will be crucial for solving the given equation.

Step 2 - Apply the Square Root Property

Using the property: z^2 = 169, we find that z = ±√169.

Step 3 - Simplify the Radical

Calculate the square root of 169: √169 = 13. Therefore, z = ±13.

Key Concepts

Solving Equations

When solving equations, especially those involving squares, it's important to recognize patterns and properties. Here, we deal with a quadratic equation because it involves a squared term, \(z^2\). The primary goal is to isolate the variable, in this case, \(z\). We can use specific methods, such as factoring, completing the square, or, in this example, the square root property. Remember, when applying these steps, we aim to simplify the equation to its most basic form and then solve for the variable.

Simplifying Radicals

Simplifying radicals involves finding the simplest form of the given radical expression. For instance, to simplify the square root of 169, we determine its prime factors or recognize it as a perfect square. Perfect squares are numbers like 1, 4, 9, 16, and so on, which result from squaring whole numbers. The square root of a perfect square is always a whole number. In our example, \(\sqrt{169}\) simplifies to 13 because 13 \(\times\) 13 = 169. Simplifying radicals makes them easier to work with and understand.

Square Roots

Square roots are fundamental in mathematics and often encountered in algebra and geometry. The square root of a number, \(b\), is a value, \(a\), such that \(a^2 = b\). For example, in our exercise, we have \(z^2 = 169\), meaning \(z\) is the square root of 169. Don't forget, when solving equations through the square root property, always consider both the positive and negative solutions. Therefore, \(\pm\) is included, giving us \(z = \pm 13\). Understanding square roots and their properties is crucial for solving such equations effectively.