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Chapter 9: Problem 18

Simplify. $$ \sqrt{25+144} $$

Short Answer

Expert verified
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Step by step solution

01

Identify the expression inside the square root

Consider the expression inside the square root: \( 25 + 144 \). These are the terms to be simplified.
02

Perform the addition inside the square root

Add the numbers inside the square root: \( 25 + 144 = 169 \).
03

Simplify the square root

Find the square root of the result: \( \sqrt{169} \). Since \( 169 \) is a perfect square, \( \sqrt{169} = 13 \).

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Key Concepts

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

square root
A square root is a mathematical function that finds the original number that was squared (multiplied by itself) to obtain the given value. For instance, the square root of 9 is 3 because 3 squared (3 x 3) equals 9.
In mathematical notation, the square root is represented as \( \sqrt{} \). When you see \( \sqrt{169} \), it means you are looking for the number which, when multiplied by itself, gives 169.
This operation undoes the effect of squaring a number, making the square root a crucial concept in algebra and geometry.
perfect squares
Perfect squares are integers that can be expressed as the product of an integer with itself. In other words, if a number can be written like \( n \times n \), it is a perfect square.
For example:
  • 4 is a perfect square because \( 2 \times 2 = 4 \).
  • 9 is a perfect square because \( 3 \times 3 = 9 \).
  • 16 is a perfect square because \( 4 \times 4 = 16 \).
Recognizing perfect squares helps to simplify square roots quickly. In our exercise, we identified that 169 is a perfect square because \( 13 \times 13 = 169 \). Thus, \( \sqrt{169} = 13 \).
addition inside radicals
When given an expression inside a square root, the first step is to simplify the expression as much as possible before taking the square root. This includes performing any addition, subtraction, multiplication, or division within the radical.
In our exercise, the expression inside the square root was \( 25 + 144 \). We combined these terms:
  • First, add 25 and 144 to get 169.
  • Next, find the square root of 169.
By simplifying inside the radical first, the problem becomes easier to manage. It鈥檚 essential to remember that addition and subtraction inside the radical must be handled before attempting to simplify the square root directly.

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