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Chapter 9: Problem 1

Find each square root. $$ \sqrt{25} $$

Short Answer

Expert verified
The square root of 25 is 5.

Step by step solution

01

Understand the Problem

We need to find the square root of 25, which means finding a number that, when multiplied by itself, gives 25.
02

Memorize or Calculate Known Squares

Recognize that 25 is a perfect square. This means it can be expressed as the square of an integer.
03

Determine the Square Root

Recall that 25 is the square of 5 because when we multiply 5 by itself (5 * 5), the result is 25. Therefore, \(\sqrt{25} = 5\).

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Key Concepts

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

Understanding Perfect Squares
Perfect squares are special numbers that are the product of an integer multiplied by itself. For example, if you take the number 5 and multiply it by itself, you get 25. Therefore, 25 is called a perfect square.

Recognizing perfect squares can simplify your mathematical problem-solving because you can quickly identify their square roots. Some commonly known perfect squares include:
  • 1 (since 1 x 1 = 1)
  • 4 (since 2 x 2 = 4)
  • 9 (since 3 x 3 = 9)
  • 16 (since 4 x 4 = 16)
  • 25 (since 5 x 5 = 25)
Knowing these values can help you solve square root problems more efficiently by recognizing them without calculation. The ability to identify perfect squares is a useful mathematical skill that aids in quicker, more confident computations.
The Role of Integers in Square Roots
Integers play a critical role when working with square roots, especially when dealing with perfect squares. If a number is a perfect square, its square root will always be an integer. Let's consider the number 25 again; its square root, 5, is an integer.

Understanding this relationship helps to streamline your mathematical process:
  • An integer is any whole number, including positive and negative numbers, as well as zero.
  • A perfect square like 36 will have a square root that is also an integer, which in this case is 6, because 6 x 6 = 36.
  • If a number is not a perfect square, its square root will likely be a decimal or fraction.
This concept helps to swiftly verify whether your calculated square root is correct. If the square root of a perfect square is not an integer, then there may be an error in your calculations.
Approaching Mathematical Problem-Solving
Problem-solving in mathematics often involves breaking down complex problems into simpler, more manageable tasks. Let's apply this to finding square roots, like in our original exercise of determining \(\sqrt{25}\).

Here鈥檚 a systematic way to approach such problems:
  • First, understand what you are being asked to solve. In this instance, finding a number that, when squared, equals 25.
  • Next, consider recognizable patterns or memorized facts such as known perfect squares. Remembering that 5 x 5 = 25 can expedite finding \(\sqrt{25}\).
  • Finally, verify your result. Ensure your proposed solution indeed satisfies the problem鈥攂y checking if 5 x 5 equals 25 in this case.
This step-by-step approach can be applied to many areas in math, reinforcing logical thinking and enhancing accuracy while reducing the likelihood of errors.

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Most popular questions from this chapter

The table shows the heat index and relative humidity for an air temperature of \(75^{\circ} \mathrm{F}\). $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Eacture ty uster } & 0 \% & 5 \% & 10 \% & 15 \% & 20 \% & 25 \% & 30 \% & 35 \% & 40 \% & 45 \% & 50 \% \\ \hline \text { Lesures }(x) & 69 & 69 & 70 & \pi & 72 & 72 & 73 & 73 & 74 & 74 & 75 \\ \hline \end{array}$$ Make a scatter plot and draw a line of fit.

ALGEBRA The measures of the angles of a triangle are in the ratio \(1: 4: 5 .\) Find the measure of each angle. (lesson \(9-3\) )

Find the distance between each pair of points. Round to the nearest tenth, if necessary.$$Q\left(5 \frac{1}{4}, 3\right), R\left(2,6 \frac{1}{2}\right)$$

Solve each equation. Round to the nearest tenth, if necessary. $$3 m^{2}=222$$

Order each set of numbers from least to greatest. $$4 . \overline{23}, 4 \frac{2}{3}, \sqrt{18}, \sqrt{16}$$
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