Tag: Prime factorization

Introduction

When a number (n) is multiplied three times, the result is known as the cube of that number. As a result, the cube of the number (n) is \({n^3}\) or n-cubed. Select the number 4 as an example. We already know that . As a result, 64 is known as the cube of 4. The cube root of a number, on the other hand, is the inverse of the cube of a number and is denoted by \(\sqrt[3]{{}}\) In the same example, 4 is called the cube root of 64. Let’s go ahead and learn more about the numbers’ cubes and cube roots.

What is a cube of a number?

A cube number is an outcome of multiplying an integer by the same integer three times. They also go by the name “perfect cubes,” or cube numbers. For example, \(4 \times 4 \times 4 = {4^3} = 64\). A  number multiplied by the same number three times is called a cube number, or a number with the exponential power of three. Because a negative number produces a negative number when it is multiplied by the same negative number three times, cube numbers of positive numbers are positive, while cube numbers of negative numbers are negative. For example, \({\left( { – 6} \right)^3}\)

In geometry, a cube’s volume is equal to the product of its length, breadth, and height. The length, breadth, and height are all equal inside the cube because it is a cube. The cube’s volume is therefore equal to , which  is its length, height, and width. This implies that a cube’s volume is a cube number.

What are the cube roots?

When we hear the word cube root, the first two words that come to mind are cube and roots. In this sense, the concepts are somewhat similar; “root” refers to the primary source of origin. As a result, we only need to consider “which number’s cube must be chosen to get the given number.”

In a nutshell, the cube root of any number is the factor that is multiplied three times to obtain the specific values. Keep in mind that a number’s cube root is the inverse of the number’s cube.

Cube roots by prime factorization

The prime factorization method can be used to calculate the cube root of a number. Begin by determining the prime factorization of a given number to find its cube root. Then, divide the obtained factors into groups, each having three identical factors. Then, to get the answer, eliminate the symbol of the cube root and multiply the factors. If any factor remains that cannot be equally divided into sets of three, the given number isn’t a perfect cube, and we cannot determine its cube root. 

Example: How to determine the cube root of 10648.

 

\(\begin{array}{l}\sqrt[3]{{10648}} = \sqrt[3]{{2 \times 2 \times 2 \times 11 \times  \times 11 \times  \times 11}}\\\;\;\;\;\;\;\;\;\;\;\; = 2 \times 11 = 22\end{array}\) 

Cube roots by estimation

If a number has several digits, it will be difficult for you to use the prime factorization and long division methods to compute the square and cube roots of the number. You will thus attempt to estimate the cube root and square root values for these numbers. Factorization becomes challenging as the number of digits rises. Estimating a square root and a cube root in such circumstances is a wise move.

The following step can be used to determine a large integer’s cube root using the estimation method if it is claimed that the provided number is a perfect cube.

Step 1: Starting with the rightmost digit, create a group of three digits starting with any cube number, such as 19683, whose cube root needs to be found.

Step 2: The unit’s digit of the necessary cube root will be provided by the first group. Since the unit digit is 3, we need to find the cube root of the cube which also has unit digit 3, i.e., \({7^3} = 343\) has unit digit 3. Hence, our unit digit for the cube root is 7.

Step 3: Now take the second group, i.e., 19, and find the cube root that is just smaller than 19 ( second group ).

\({2^3} < 19 < {3^3}\).

8 or  is just smaller than 19 hence our tenths place digit is 2.

Hence, the cube root of 19683 is equal to

\(\sqrt {19683}  = 27\)

Solved examples

Example 1: Determine the following:

1. \(\sqrt[3]{{729}}\)

Solution: We will be using the prime factorization method.

From it, we will get

Now operate both the sides under the cube root.

\(\begin{array}{l}\sqrt[3]{{729}} = \sqrt[3]{{3 \times 3 \times 3 \times 3 \times 3 \times 3}}\\\;\sqrt[3]{{729}} = 3 \times 3 = 9\end{array}\)

2. \(\sqrt[3]{{1728}}\)

Solution: We will be using the prime factorization method.

From it, we will get

Now operate both the sides under the cube root.

\(\begin{array}{l}\sqrt[3]{{1728}} = \sqrt[3]{{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3}}\\\;\sqrt[3]{{1728}} = 2 \times 2 \times 3 = 12\end{array}\)

3. \(\sqrt[3]{{2744}}\)

Solution: We will be using the prime factorization method.

From it, we will get

Now operate both the sides under the cube root.

\(\begin{array}{l}\sqrt[3]{{2744}} = \sqrt[3]{{2 \times 2 \times 2 \times 7 \times 7 \times 7}}\\\;\sqrt[3]{{2744}}\; = 2 \times 7 = 14\end{array}\)

Example 2: Determine the cube root of 59319 using estimation.

Solution:

The given number is 59319

First, we have to make the group of three from the right side. So we have the first group (319) and the second group (59).

To find the unit digit of the cube root, we have to find the cube that has the unit digit equal to the unit digit of the first group.

\({9^3} = 729\)

The cube of 9 has the unit digit 9, so our unit digit of the cube root is 9.

Now take the second group 59 and find the cube root that is just smaller than 59.

\({3^3} < 59 < {4^3}\).

27 or \({3^3}\) is just smaller than 59, hence our tenth place digit is 3.

Hence, the cube root of 59319 is equal to

\(\sqrt {59319}  = 39\)

Summary

When we hear the word cube root, the first two words that come to mind are cube and roots. In this sense, the concepts are somewhat similar; “root” refers to the primary source of origin. As a result, we only need to consider “which number’s cube must be chosen to get the given number.” The cube root of any number is the factor that is multiplied three times to obtain the specific values. Keep in mind that a number’s cube root is the inverse of the number’s cube.

When we say n is a cube root of m, then we can denote it as \(\sqrt[3]{m} = n\), with a small 3 written on the top left of the sign. The radical sign \(\sqrt[3]{{}}\)  is used as a cube root symbol for any number. Another way to express cube root is to write 1/3 as a number’s exponent. In cube roots, negative values are allowed, unlike in square roots,

For example, \(\sqrt[3]{{ – 216}} =  – 6\)

If you’re struggling with the concept of finding the cube root of a number, don’t worry – we’re here to help! Check out our video lesson no. 15 in 88guru’s online Math tuition for 8th class students.

Frequently Asked Questions

1. State the definition of cube root.

Ans: The cube root of any number is the factor that is multiplied three times to obtain the specific values. Keep in mind that a number’s cube root is the inverse of the number’s cube.

2. How is a cube root different from a square root?

Ans: A cube root is a number that when cubed yields the wanted number, whereas a square root yields the wanted number when squared. Furthermore, in cube roots, negative values are allowed, unlike in square roots.

3. What is the name of the method used for finding cube roots?

Ans: The prime factorization method is used to calculate the cube root of a number.

4. What is the cube root of 125?

Ans: 125 is a perfect cube, and 5 is the cube root of 125.

5. How Can a Cube Root Be Simplified?

Ans: The prime factorization method can be used to simplify the cube root. First, prime factorize the given number and then extract the common factors in groups of three. To find the answer, multiply these common factors by two.

Introduction

The relationship between a number and its square root is important to understand. Squaring a number means multiplying it by itself, while finding the square root of a positive number involves finding the number that, when squared, results in the original number. For example, if the square of a number p is q, then the square root of q is equal to p. 

As an example, 2 squared is 4, and the square root of 4 is either +2 or -2.

What is a square of a number?

The square of a number is defined as the product of the number with the number itself. This is represented by a superscript 2 in front of the number. Here are a few examples.

• squared is \(2 \times 2 = {2^2} = 4\)
• 3 square is \(3 \times 3 = {3^2} = 9\)
• 4 square is \(4 \times 4 = {4^2} = 16\)

Square roots

What are square roots?

The square root operation is the inverse of squaring a number. Basically, it is the number which, when multiplied by itself, gives the number whose square root we are trying to find. The square root is operated by the radical symbol \(\sqrt {} \) and the number inside the radical is called the radicand.

Hence, given a number , its square root is represented by \(\sqrt p \)  and if we square \(\sqrt p \) we arrive back at the original number. Thus,

\(\sqrt p  \times \sqrt p  = \sqrt {{p^2}}  = p\)

Square root symbol

The square root function, which is a one-one function, takes a positive number and yields its square root. If it is provided a negative number, the answer is complex.

\(f\left( a \right) = \sqrt a \)

Example: The square root of 4 is equal to \(\sqrt 4  =  \pm 2\)

The square root of 9 is equal to = \(\sqrt 9  =  \pm 3\)

The square root of 16 is equal to = \(\sqrt {16}  =  \pm 4\).

Square roots by prime factorization

We can find the square root of a number by using the prime factorisation method.

1. We start by finding out the prime factors of the number.
2. We then group the same factors into pairs of two.
3. We then take one number each from these pairs and multiply them together. The product we thus obtain is the square root we need.

Example: Find the square root of 36 by using the prime factorization method.

Upon factorising 36, we see that the prime factors come out to be

\(36 = 2 \times 2 \times 3 \times 3\)

We now group the same numbers together

\(36 = \left( {2 \times 2} \right) \times \left( {3 \times 3} \right)\)

And then take one number from each pair. The product thus obtained is the square root. Hence

\(36 = {\left( {2 \times 3} \right)^2} = {6^2}\)

Therefore, the square root of 36 is \( \pm 6\).

Square roots by estimation

In various cases, the perfect square root of a number doesn’t exist and we estimate the nearest value. This can be explained by taking the example of 24.

1. Start by finding the nearest perfect squares to 24. These are 25 and 16, whose square roots are 4 and 5.
2. Since \({5^2}\)  is closer to 24, we increase 4 to 4.5 and check again.
3. Again, \({4^2} = 20.25,So\;{5^2}\) is still closer to 24. We adjust again and go from 4.5 to 4.8.
4. This way, we continue till a close enough estimate has been made. In this case, it comes out to be 4.8989

Also read: How to Find Cube Root of a Number

Square roots by Long division method

It is a method used to find the square roots of large numbers by dividing them into parts and getting the exact value of the square root of the number.

Example: Find the square root of 150 using the long division method?

To obtain the square root of a number using long division:

1. Group the digits of the number into pairs starting from the rightmost digit. This is done by placing bars on top of the groups for easy identification.
2. Find the largest perfect square less than or equal to the leftmost pair (which is 1 here) and use its square root as the divisor and quotient (both come out to be 1). Divide the leftmost pair (1 here) by the divisor, and bring down the next pair of digits (50 here). 
3. Bring the last digit of the quotient (1 here) to the divisor (1 again) and using the sum of these two numbers (1+1=2). Now try to find the largest two digit number that starts with 2 and doubling which, will give us a number below the number we have brought down (50 here).
4. Continue the process in this way and add a decimal point in the quotient and zeroes in the dividend if required untill a desired accuracy is reached. The final quotient will become our square root.

The following table better demonstrates this:

Hence, the square root of 150 is equal to 12.247

Solved Examples

1. Find the square root of 144 via prime factorization.

When we factorise 144, we arrive at the following equation

\(144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\)

We group these factors into pairs

\(144 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {3 \times 3} \right)\)

Then take one factor from each pair.

\(144 = 2 \times 2 \times 3\)

Hence, 12 is the square root of 144.

2. Estimate the square root of 30.

The nearest perfect squares are 25 and 36 and therefore, the square root of 30 must lie between 5 and 6. Since \({5.5^2} = 30.25\), the square root of 30 is very close to 5.5. We now try the guess 5.4, whose square is 29.16. To get better accuracy, let us go up to 5.45, whose square is 29.7025. Continuing this way, we get the desired answer as 5.4772.

Summary

This article discussed the following concepts:

1. What is meant by the square and square root of a number?
2. How square roots are represented.
3. Various ways of finding square roots of a number. These included prime factorization, estimation, and long division methods.

 

Frequently Asked Questions

1.What are the methods used in finding the square root of a decimal number?

The square root of a decimal number may be found via long-division or estimation methods.

2. When is prime factorization useful?

Prime factorization is only useful when the given number is a perfect square. Otherwise, it fails.

3. What are perfect squares?

Perfect squares are those numbers whose square roots are whole numbers, rather than decimal numbers.

4. How do we find square roots of numbers which aren’t perfect squares?

Estimation and long division methods can aid us in such scenarios.

Introduction

In every area of our lives, we are surrounded by numbers. Factors and multiples are also commonly used in our daily life. We use factors when we want to arrange things differently. For example, arrange books in rows and columns, group children in different ways, etc. Let’s consider the number 112. 112 can be divided by 1, 2, 4, 7, 8, 14, 16, 28, 56, and 112. So the factors of 112 are 1, 2, 4, 8, 14, 16, 28, 56 and 112. Likewise, the factors of 112 are -1, -2, -4, -8, -14, -16, -28, -56, and -112. Therefore, when looking for or solving problems involving factors, only positive numbers, negative numbers, whole numbers, and non-decimal numbers are considered.

Properties of factors

• All integers have a finite number of factors.
• The factor of a number is always less than or equal to the number; it can never be greater than the number.
• Except 0 and 1, every integer has at least two factors: 1 and the number itself.
• Find factors by using division and multiplication.

Prime factorization

When we write a number as the product of all its prime factors, it is called prime factorization. Every number in prime factorization is prime. To write a number as the product of prime factors, we may sometimes have to repeat these factors as well.

Example: To write the prime factorization of 8, we can write

\(8{\rm{ }} = \;2 \times 2 \times 2\)

that is, the prime factor 2 repeated 3 times. To write the prime factorization of 112, we can write

\(112 = 2 \times 2 \times 2 \times 2 \times 7\),

which is the prime factor of 2 repeated four times and multiplied by 7.

Therefore, the prime factors of 112 are 2 and 7. A number with more than two factors is called a composite number.

The number 112 has more than two factors.

Therefore, 112 is a composite number.

we know,

A number that is not divisible by any other number is called a prime number.

In the factors of 112, we get that

\({2^4}{\rm{ \times }}7\)  is prime.

Steps to find factors of 112

1. First we divide the given number 112 by to get a remainder of 0.
2. After getting the answer, all the numbers we get are called factors of the given number.
3. 1 is the only number that has factors of all numbers.
4. The given number itself is the highest factor of the given number.

What are the factors of 112?

We use LCM to find the prime factorization of 112.

Prime factors of \(112{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}7\)

Prime factor of \(112 = {2^4}{\rm{ \times }}7\)

112 has 10 factors, namely\(1,{\rm{ }}2,{\rm{ }}4,{\rm{ }}7,{\rm{ }}8,{\rm{ }}14,{\rm{ }}16,{\rm{ }}28,{\rm{ }}56{\rm{ }}and{\rm{ }}112\) .

All factors of \(112{\rm{ }} = {\rm{ }}1,{\rm{ }}2,{\rm{ }}4,{\rm{ }}7,{\rm{ }}8,{\rm{ }}14,{\rm{ }}16,{\rm{ }}28,{\rm{ }}56,{\rm{ }}and{\rm{ }}112\)

Factors of 112 in pairs

When we make pairs of factors, the product of the two factors is the given number itself.

Factor pairs of \(112:{\rm{ }}\left( {1 \times 112} \right),{\rm{ }}\left( {2 \times 56} \right),{\rm{ }}\left( {4 \times 28} \right),{\rm{ }}\left( {7 \times 16} \right),{\rm{ }}\left( {8 \times 14} \right)\)

112 factor pairs of \(\left( {1,{\rm{ }}112} \right),{\rm{ }}\left( {2,{\rm{ }}56} \right),{\rm{ }}\left( {4,28} \right),{\rm{ }}\left( {7,{\rm{ }}16} \right),{\rm{ }}\left( {8,14} \right)\)

We know the multiplication property,

\(a{\rm{ }} \times {\rm{ }}b{\rm{ }} = {\rm{ }}b{\rm{ }} \times {\rm{ }}a\)

Pairwise factors of \(112:{\rm{ }}\left( {1,{\rm{ }}112} \right),{\rm{ }}\left( {2,{\rm{ }}56} \right),{\rm{ }}\left( {4,{\rm{ }}28} \right),{\rm{ }}\left( {7,{\rm{ }}16} \right),{\rm{ }}\left( {8,{\rm{ }}14} \right)\)

The divisors of \(112:{\rm{ }}\left( {1,{\rm{ }}112} \right),{\rm{ }}\left( {2,{\rm{ }}56} \right),{\rm{ }}\left( {4,{\rm{ }}28} \right),{\rm{ }}\left( {7,{\rm{ }}16} \right),{\rm{ }}\left( {8,{\rm{ }}14} \right)\) are the same.

Factor tree of 112

A factor tree is a special graph where we find the factors of a number and then find the factors of those numbers until we can no longer factor them. In the end, all we get are the prime factors of the original numbers.

A factor tree of 112 is a list of prime numbers when multiplied by the original number 112.

\(\begin{array}{*{20}{l}}{112{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}56}\\{56{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}28}\\{28{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}14}\\{14{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}7}\end{array}\)

If we write the multiple it will be \(112{\rm{ }} \times {\rm{ }}2\)

When splitting 56 further and writing it as a multiple of the number, it would be \(28{\rm{ }} \times {\rm{ }}2.\)

When splitting 28 further and writing it as a multiple of the number, it would be \(14{\rm{ }} \times {\rm{ }}2.\)

When splitting 14 further and writing it as a multiple of the number, it would be \(7{\rm{ }} \times {\rm{ }}2.\)

In prime factors the sum of this number is \(2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}7\).

So,

The factors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56 and 112 itself.

Did you know

1.  Except 0 and 1, every integer has at least two factors: 1 and the number itself.
2. The factor of a number is always less than or equal to the number; it can never be greater than the number.112 is not a perfect square..
3. 112 is a composite number.
4. 112 is an even number.
5. 112 is not a perfect square.

Solved example

1. Find all the factors of 20.

Step 1: Write down all the numbers from 1 to 20.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Step 2: Now check which of these numbers are divisible by 20 and have no remainder.\(\begin{array}{*{20}{l}}{\;\;\;\;\;\;\;\;\;20/1{\rm{ }} = {\rm{ }}20}\\{\;\;\;\;\;\;\;\;\;20/2{\rm{ }} = {\rm{ }}10}\end{array}\;\;\;\;\;\;\;\;\;20/3{\rm{ }} = {\rm{ }}indivisible\) .

Go ahead and divide 20 by these numbers.

Step 3: The factors of 20 are 1, 2, 4, 5, 10, and 20.

2. Find all the factors of 31.

31 is a prime number. The only two numbers that divide 31 are 1 and 31.

So the factors of 31 are 1 and 31.

3. Find the prime factors of 144.

As the name suggests, prime factorization is a method of deriving the prime factors of any number. Prime factors are prime numbers. The factors of these numbers are 1 and the numbers themselves. For example, 13 is a prime number because the factors of this number are 1 and 13.

Consider the number 144. Consider first the smallest possible factor, which is 2.

\(144{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}72{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}36{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}18{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}9{\rm{ }} = {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}3{\rm{ }} \times {\rm{ }}3\) So, the prime factors of 144 are 2 and 3 because these factors are prime numbers.

Conclusion

The easiest way to determine the factors of a number is to divide by the smallest prime number with no remainder and continue the process. 112 has more than two factors, namely 1, 2, 4, 7, 8, 14, 16, 28, 56 and 112, so it is a composite number. 112 is an even number, and it is not a perfect square.

 

Frequently asked question 

1. What are the factors of 112?

Since 112 is a composite number, it has more than 2 factors, so the factors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56, and 112.

2. What is the prime factorization of 112?

Prime factorization of 112, we can write \(112 = \;2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times {\rm{ }}2{\rm{ }} \times \;7\)

3. What are the negative pair factors of 112?

The negative pair factors of 112 are \(\left( { – 1,{\rm{ }} – 112} \right),{\rm{ }}\left( { – 2,{\rm{ }} – 56} \right),{\rm{ }}\left( { – 4,{\rm{ }} – 28} \right),{\rm{ }}\left( { – 7,{\rm{ }} – 16} \right)\) and \(( – 8, – 14).\)