Problem 11 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(1-12,\) or indicate that the root is not a real number. $$\sqrt{(-13)^{2}}$$

Step-by-Step Solution

Answer

The value of the expression \( \sqrt{(-13)^{2}} \) is 13.

Step 1: Square the number inside the parentheses

In the given expression \( \sqrt{(-13)^{2}} \), the number -13 must be squared first. Squaring any number involves multiplying it by itself. Thus, \((-13) * (-13) = 169.\)

Step 2: Take the square root of the squared number

The next step is to take the square root of 169, which we obtained after squaring -13. The square root of 169 is 13.

Step 3: Verification

To verify the solution, square the result obtained in step 2. Squaring 13 returns 169, which confirms the solution is correct.

Key Concepts

Understanding Real Numbers

Real numbers are numbers that can be found on the number line. They include both rational numbers, like fractions or whole numbers, and irrational numbers, which cannot be expressed as exact fractions.

• Rational numbers are numbers that can be written as a fraction where both the numerator and denominator are integers.
• Irrational numbers are numbers that cannot be written in a simple fraction form.

Real numbers play a crucial role in algebra exercises. In our exercise, \\( (-13)^2 \), both -13 and the result 169 are real numbers. When evaluating an expression, it is essential to ensure that the result qualifies as a real number. This means that when you do operations like squaring or taking square roots, the outcomes should still be within the set of real numbers.

Explaining Square Roots

The square root of a number is one of the two equal factors of that number. For instance, the square root of 169 is 13 because \\( 13 \times 13 = 169 \). Taking the square root is essentially the opposite operation of squaring a number.

• Every positive real number has two square roots: a positive and a negative one. For example, both 13 and -13 are square roots of 169.
• The principal square root is normally the positive number.

In algebra exercises, determining the square root helps in simplifying expressions. For the problem \( \sqrt{(-13)^{2}} \), after squaring -13 to get 169, you take the square root of 169, resulting in 13. Always remember, when dealing with square roots, you are essentially finding the number which can multiply by itself to give the original number.

Squaring Numbers in Algebra

Squaring a number means multiplying it by itself. This is a fundamental operation in algebra and appears in many exercises.

• If a number is positive, its square is positive.
• If a number is negative, its square is still positive because multiplying two negative numbers results in a positive product.

In the given exercise, you square the number -13. Squaring -13 means calculating \\( (-13) \times (-13) = 169 \). This process confirms that no matter the signs of the original number, squaring will provide a non-negative result. Understanding square numbers is essential as it helps simplify expressions and check accuracy in solutions.