 Square root of 225 definition The square root of 225 in mathematical form is written with the radical sign like this √225. We call this the square root of 225 in radical form. The square root of 225 is a quantity (q) that when multiplied by itself will equal 225. √225 = q × q = q2 Show Is 225 a perfect square? 225 is a perfect square if the square root of 225 equals a whole number. As we have calculated further down on this page, the square root of 225 is a whole number. 225 is a perfect square. Is the square root of 225 rational or irrational? The square root of 225 is a rational number if 225 is a perfect square. It is an irrational number if it is not a perfect square. Since 225 is a perfect square, it is rational number. This means that the answer to "the square root of 225?" will have no decimals. √225 is a rational number Can the square root of 225 be simplified? A square root of a perfect square can be simplified because the square root of a perfect square will equal a whole number: √225 = 15 How to calculate the square root of 225 with a calculator The easiest and most boring way to calculate the square root of 225 is to use your calculator! Simply type in 225 followed by √x to get the answer. We did that with our calculator and got the following answer: √225 = 15 How to calculate the square root of 225 with a computer If you are using a computer that has Excel or Numbers, then you can enter SQRT(225) in a cell to get the square root of 225. Below is the result we got: SQRT(225) = 15 What is the square root of 225 written with an exponent? All square roots can be converted to a number (base) with a fractional exponent. The square root of 225 is no exception. Here is the rule and the answer to "the square root of 225 converted to a base with an exponent?": √b = b½ √225 = 225½ How to find the square root of 225 by long division method Here we will show you how to calculate the square root of 225 using the long division method. This is the lost art of how they calculated the square root of 225 by hand before modern technology was invented. Step 1) Set up 225 in pairs of two digits from right to left:Step 2) Starting with the first set: the largest perfect square less than or equal to 2 is 1, and the square root of 1 is 1. Therefore, put 1 on top and 1 at the bottom like this: Step 3) Calculate 2 minus 1 and put the difference below. Then move down the next set of numbers. Step 4) Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem: 2? × ? ≤ 125 The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 5. Replace the question marks in the problem with 5 to get: 25 × 5 = 125. Now, enter 5 on top, and 125 at the bottom: The difference between the bottom two numbers is zero, therefore, you are done! The answer is the green numbers on top. Once again, the square root of 225 is 15.Square Root of a Number Please enter another number in the box below to get the square root of the number and other detailed information like you got for 225 on this page. Notes Remember that negative times negative equals positive. Thus, the square root of 225 does not only have the positive answer that we have explained above, but also the negative counterpart. We often refer to perfect square roots on this page. You may want to use the list of perfect squares for reference. Square Root of 226 Here is the next number on our list that we have equally detailed square root information about. Copyright | Privacy Policy | Disclaimer | Contact The square root of the number 225 is 15, making it a perfect square as 15 is an integer. Mathematically, we can write it as √225 = 15. The square root of 225 equal to 15 is a rational number, which means it is a ratio of two integers. In this article, we will analyze and find the square root of 225 using various mathematical techniques, such as the approximation method and the long division method. What Is the Square Root Of 225?The square root of the number 225 is 15. The square root can be defined as the quantity that can be doubled to produce the square of that similar quantity. In simple words, it can be explained as: √225 = √(15 x 15) √225 = √(15)$^2$ √225 = ±15 The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 15. Hence 15 is 225’s square root. The square root generates both positive and negative integers. How To Calculate the Square Root of 225?You can calculate the square root of 225 using any of two vastly used techniques in mathematics; one is the Approximation technique, and the other is the Long Division method. The symbol √ is interpreted as 225 raised to the power 1/2. So any number, when multiplied by itself, produces its square, and when the square root of any squared number is taken, it produces the actual number. Let us discuss each of them to understand the concepts better. Square Root by Long Division MethodThe process of long division is one of the most common methods used to find the square roots of a given number. It is easy to comprehend and provides more reliable and accurate answers. The long division method reduces a multi-digit number to its equal parts. Learning how to find the square root of a number is easy with the long division method. All you need are five primary operations- divide, multiply, subtract, bring down or raise, then repeat. Following are the simple steps that must be followed to find the square root of 225 using the long division method: Step 1First, write the number 225 in the division symbol, as shown in figure 1. Step 2Starting from the right side of the number, divide the number 225 into pairs such as 25 and 2. Step 3Now divide the digit 25 by a number, giving a number either 25 or less than 25. Therefore, in this case, the remainder is 1, whereas the quotient is 1. Step 4After this, bring down the next pair, 25. Now the dividend is 125. To find the next divisor, we need to double our quotient obtained before. Doubling 1 gives 2; hence consider it as the next divisor. Step 5Now pair 2 with another number to make a new divisor that results in $\leq$ 125 when multiplied with the divisor. Step 6Adding 5 to the divisor and multiplying 25 with 5 results in 125 $\leq$ 125. The remainder obtained is 0. Step 7The resulting quotient 15 is the square root of 225. Figure 1 given below shows the long division process in detail: Figure 1 Important points
Is Square Root of 225 a Perfect Square?The number 225 is a perfect square. A number is a perfect square if it splits into two equal parts or identical whole numbers. If a number is a perfect square, it is also rational. A number expressed in p/q form is called a rational number. All the natural numbers are rational. A square root of a perfect square is a whole number; therefore, a perfect square is a rational number. A number that is not a perfect square is irrational as it is a decimal number. As far as 225 is concerned, it is a perfect square. It can be proved as below: Factorization of 225 results in 15 x 15, which can also be expressed as 15$^2$. Taking the square root of the above expression gives: = √(15$^2$) = (15$^2$)$^{1/2}$ = 15 This shows that 225 is a perfect square and a rational number. Therefore the above discussion proves that the square root of 225 is equivalent to 15. Images/mathematical drawings are created with GeoGebra. Square Roots List By the end of this section, you will be able to:
Before you get started, take this readiness quiz.
Remember that when a number n is multiplied by itself, we write \(n^2\) and read it “n squared.” For example, \(15^2\) reads as “15 squared,” and 225 is called the square of 15, since \(15^2=225\).
If \(n^2=m\), then m is the square of n. Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is m is called a square root of m.
If \(n^2=m\), then n is a square root of m. Notice \((−15)^2=225\) also, so −15 is also a square root of 225. Therefore, both 15 and −15 are square roots of 225. So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, \(\sqrt{m}\), denotes the positive square root. The positive square root is also called the principal square root. We also use the radical sign for the square root of zero. Because \(0^2=0\), \(\sqrt{0}=0\). Notice that zero has only one square root.
\(\sqrt{m}\) is read as “the square root of m.” If \(m=n^2\), then \(\sqrt{m}=n\), for \(n \ge 0\). The square root of m, \(\sqrt{m}\), is the positive number whose square is m. Since 15 is the positive square root of 225, we write \(\sqrt{225}=15\). Fill in Figure to make a table of square roots you can refer to as you work this chapter. We know that every positive number has two square roots and the radical sign indicates the positive one. We write \(\sqrt{225}=15\). If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \(−\sqrt{225}=−15\).
Simplify:
1. \[\begin{array}{ll} {}&{\sqrt{36}}\\ {\text{Since} 6^2=36}&{6}\\ \end{array}\] 2. \[\begin{array}{ll} {}&{\sqrt{196}}\\ {\text{Since} 14^2=196}&{14}\\ \end{array}\] 3. \[\begin{array}{ll} {}&{−\sqrt{81}}\\ {\text{The negative is in front of the radical sign}}&{−9}\\ \end{array}\] 4. \[\begin{array}{ll} {}&{−\sqrt{289}}\\ {\text{The negative is in front of the radical sign}}&{−17}\\ \end{array}\]
Simplify:
implify:
Simplify:
1. \[\begin{array}{ll} {}&{\sqrt{−169}}\\ {\text{There is no real number whose square is} s−169}&{\sqrt{−169} \text{is not a real number.}}\\ \end{array}\] 2. \[\begin{array}{ll} {}&{−\sqrt{64}}\\ {\text{The negative is in front of the radical sign}}&{−8}\\ \end{array}\]
Simplify:
Simplify:
When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.
Simplify:
1. \[\begin{array}{ll} {}&{\sqrt{25} +\sqrt{144}}\\ {\text{Use the order of operations}}&{5+12}\\ {\text{Simplify.}}&{17}\\ \end{array}\] 2. \[\begin{array}{ll} {}&{\sqrt{25+144}}\\ {\text{Simplify under the radical sign.}}&{\sqrt{169}}\\ {\text{Simplify.}}&{13}\\ \end{array}\] Notice the different answers in parts 1 and 2!
Simplify:
Simplify:
So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at Table below.
The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that \(\sqrt{5}\) must be between 2 and 3. Using inequality symbols, we write: \(2<\sqrt{5}<3\)
Estimate \(\sqrt{60}\) between two consecutive whole numbers. AnswerThink of the perfect square numbers closest to 60. Make a small table of these perfect squares and their squares roots.
Estimate the square root \(\sqrt{38}\) between two consecutive whole numbers. Answer\(6<\sqrt{38}<7\)
Estimate the square root \(\sqrt{84}\) between two consecutive whole numbers. Answer\(9<\sqrt{84}<10\)
There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the \(\sqrt{x}\) key on your calculator. You will use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is \(\approx\) and it is read ‘approximately.’ Suppose your calculator has a 10-digit display. You would see that \(\sqrt{5} \approx 2.236067978\) If we wanted to round \(\sqrt{5}\) to two decimal places, we would say \(\sqrt{5} \approx 2.24\) How do we know these values are approximations and not the exact values? Look at what happens when we square them: \[\begin{array}{c} {(2.236067978)^2=5.000000002}\\ {(2.24)^2=5.0176}\\ \end{array}\] Their squares are close to 5, but are not exactly equal to 5. Using the square root key on a calculator and then rounding to two decimal places, we can find: \[\begin{array}{c} {\sqrt{4}=2}\\ {\sqrt{5} \approx 2.24}\\ {\sqrt{6} \approx 2.45}\\ {\sqrt{7} \approx 2.65}\\ {\sqrt{8} \approx 2.83}\\ {\sqrt{9}=3}\\ \end{array}\]
Round \(\sqrt{17}\) to two decimal places. Answer\[\begin{array}{ll} {}&{\sqrt{17}}\\ {\text{Use the calculator square root key.}}&{4.123105626...}\\ {\text{Round to two decimal places.}}&{4.12}\\ {}&{\sqrt{17} \approx 4.12} \end{array}\]
Round \(\sqrt{11}\) to two decimal places. Answer\(\approx 3.32\)
Round \(\sqrt{13}\) to two decimal places. Answer\(\approx 3.61\)
What if we have to find a square root of an expression with a variable? Consider \(\sqrt{9x^2}\). Can you think of an expression whose square is \(9x^2\)? \[\begin{array}{cc} {(?)^2=9x^2}&{}\\ {(3x)^2=9x^2}&{\text{so} \sqrt{9x^2}=3x}\\ \end{array}\] When we use the radical sign to take the square root of a variable expression, we should specify that x≥0x≥0 to make sure we get the principal square root. However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number and so we will not write \(x \ge 0\) next to every radical. What about square roots of higher powers of variables? Think about the Power Property of Exponents we used in Chapter 6. \((a^m)^n=a^{m·n}\) If we square \(a^m\), the exponent will become 2m. \((a^m)^2=a^{2m}\) How does this help us take square roots? Let’s look at a few: \[\begin{array}{cc} {\sqrt{25u^8}=5u^4}&{\text{Because} (5u^4)^2=25u^8}\\ {\sqrt{16r^{20}}=4r^{10}}&{\text{Because} (4r^{10})^2=16r^{20}}\\ {\sqrt{196q^{36}}=14q^{18}}&{\text{Because} (14r^{18})^2=196q^{36}}\\ \end{array}\]
Simplify:
1. \[\begin{array}{ll} {}&{\sqrt{x^6}}\\ {\text{Since} (x^3)^2=x^6}&{x^3}\\ \end{array}\] 2. \[\begin{array}{ll} {}&{\sqrt{y^{16}}}\\ {\text{Since} (y^8)^2=y^{16}}&{y^8}\\ \end{array}\]
Simplify:
Simplify:
Simplify: \(\sqrt{16n^2}\) Answer\[\begin{array}{ll} {}&{\sqrt{16n^2}}\\ {\text{Since} (4n)^2=16n^2}&{4n}\\ \end{array}\]
Simplify: \(\sqrt{64x^2}\). Answer\(8x\)
Simplify: \(\sqrt{169y^2}\). Answer\(13y\)
Simplify: \(−\sqrt{81c^2}\). \[\begin{array}{ll} {}&{−\sqrt{81c^2}}\\ {\text{Since} (9c)^2=81c^2}&{−9c}\\ \end{array}\]
Simplify: \(−\sqrt{121y^2}\). Answer\(−11y\)
Simplify: \(−\sqrt{100p^2}\). Answer\(−10p\)
Simplify: \(\sqrt{36x^{2}y^{2}}\). Answer\[\begin{array}{ll} {}&{\sqrt{36x^{2}y^{2}}}\\ {\text{Since} (6xy)^2=\sqrt{36x^{2}y^{2}}}&{6xy}\\ \end{array}\]
Simplify: \(\sqrt{100a^{2}b^{2}}\). Answer10ab
Simplify: \(\sqrt{225m^{2}n^{2}}\). Answer15mn
Simplify: \(\sqrt{64p^{64}}\). Answer\[\begin{array}{ll} {}&{\sqrt{64p^{64}}}\\ {\text{Since} (8p^8)^2=\sqrt{64p^{64}}}&{8p^8}\\ \end{array}\]
Simplify: \(\sqrt{49x^{30}}\). Answer\(7x^{15}\)
Simplify: \(\sqrt{81w^{36}}\) Answer\(9w^{18}\)
Simplify: \(\sqrt{121a^{6}b^{8}}\) Answer\[\begin{array}{ll} {}&{\sqrt{121a^{6}b^{8}}}\\ {\text{Since} (11a^{3}b^{4})^2=\sqrt{121a^{6}b^{8}}}&{11a^{3}b^{4}}\\ \end{array}\]
Simplify: \(\sqrt{169x^{10}y^{14}}\) Answer\(13x^{5}y^{7}\)
Simplify: \(\sqrt{144p^{12}q^{20}}\) Answer\(\sqrt{12p^{6}q^{10}}\) Access this online resource for additional instruction and practice with square roots. Key Concepts
square of a number
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What are the square roots of 225?
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