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Chapter 0: Problem 8

Evaluate each expression or indicate that the root is not a real number. $$\sqrt{144+25}$$

Short Answer

Expert verified
\(\sqrt{144+25} = 13\)

Step by step solution

01

Addition

Add together the two numbers under the square root: \(144 + 25 = 169\).
02

Calculate the Square Root

Next, calculate the square root of the result from the first step: \(\sqrt{169} = 13\).

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Key Concepts

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

Understanding Real Numbers
Real numbers encompass all the numbers that can be found on the number line. This includes both rational numbers, like 2 and 3/4, and irrational numbers, like \( \sqrt{2} \) and \( \pi \).
When dealing with square roots, we are often looking for numbers that multiply by themselves to give a particular number.
In the expression \( \sqrt{169} \), 13 is a real number because \( 13 \times 13 = 169 \).
  • Real numbers can be negative, positive, or zero.
  • They are used every day in various calculations, from banking to engineering.
  • Understanding real numbers helps us navigate not only simple mathematics but complex equations as well.
If a square root results in a non-real number (like the square root of a negative), it doesn't appear on the real number line.
The Role of Addition in Expressions
Addition is one of the fundamental operations in mathematics. It's all about bringing numbers together to form a new total.
In our example, we added 144 and 25 to get 169.
This process is crucial because you often need to perform operations inside parentheses or under a square root symbol first when evaluating expressions.
  • Addition is associative, meaning \( (a + b) + c = a + (b + c) \).
  • It's commutative, so \( a + b = b + a \).
  • This makes it easy to handle various calculations smoothly.
By mastering addition, you will simplify your workflow when dealing with complex mathematical expressions.
Evaluating Mathematical Expressions
Evaluating expressions means finding the numerical value of a combination of numbers and operations. It's like unraveling a mystery by following specific steps.
When you face an expression like \( \sqrt{144 + 25} \), begin by following the order of operations (often remembered by PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
  • First, perform operations inside parentheses or brackets.
  • Compute any exponents or roots next, which we did by finding \( \sqrt{169} \).
  • By following the correct sequence, you ensure the accuracy of your results.
Understanding how to break down expressions and perform operations step by step is key to mastering mathematics.

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Most popular questions from this chapter

Factor completely. $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \cdot \frac{8}{15}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Solve each equation. $$\frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6}$$

Factor completely. $$-x^{2}-4 x+5$$
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