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Chapter 9: Problem 12

Solve equation by using the square root property. Simplify all radicals. \(z^{2}=169\)

Short Answer

Expert verified
z = 卤13

Step by step solution

01

Identify the equation

The given equation is \(z^{2}=169\).
02

Apply the square root property

To solve the equation \(z^{2}=169\), take the square root of both sides. This gives: \( z = \pm \sqrt{169} \)
03

Simplify the radicals

Simplify the square root of 169. We know that \( \sqrt{169} = 13 \). Therefore, \( z = \pm 13 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Quadratic Equations
Quadratic equations are mathematical expressions of the form ax^2 + bx + c = 0. In our exercise, we work with a specific type of quadratic equation where the linear term (bx) and the constant term (c) are zero. z^2 = 169 is an example of such a simple quadratic equation.

To solve a quadratic equation using the square root property, follow these steps:
  • Isolate the squared term (z^2).
  • Apply the square root to both sides of the equation.
  • Simplify the radicals, if possible.
  • Don't forget to consider both the positive and negative roots.

Understanding these steps can help you solve more complex quadratics in the future, even when they aren't as straightforward as this one!
Simplifying Radicals
Simplifying radicals is the process of reducing the expression under the square root sign to its simplest form. When you encounter a square root, look for perfect squares: numbers that are squares of integers like 4, 9, 16, and 169.

In our exercise, we work with \( \sqrt{169} = 13 \).To simplify this radical:
  • Identify the perfect square under the radical sign.
  • Take the square root of that number.
  • Replace the radical with the simplified result.

This makes solving equations much easier since you'll be working with simpler numbers!
Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 169 is 13 because \(13 \times 13 = 169\).

Square roots are fundamental in many areas of mathematics. When solving equations using the square root property, remember that every positive number has two square roots: a positive root and a negative root. So for the equation \( z^2 = 169 \), the solutions are \( z = 13 \) and \( z = -13 \).

Square roots are symbolized by \( \sqrt{} \) (radical sign). Always be careful in handling both positive and negative solutions to avoid missing any possible answer!

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ 2 x^{2}=5-2 x $$

Solve each quadratic equation for complex solutions by the square root property, with \(k<0 .\) Write solutions in standard form. $$ (3 x+2)^{2}=-18 $$

Use a calculator with a square root key to solve each equation. Round your answers to the nearest hundredth. \((k+2.14)^{2}=5.46\)

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 3 x^{2}-2 x+3=0 $$

Write each quotient in standard form. $$ \frac{40}{2+6 i} $$
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