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ClassDojo helps teachers, parents, and students build amazing classroom communities.

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Suppose we have an expression given by z2 = y. We can say the square root of a number y is a number such that the above equation is satisfied.

Thus, when a number z is multiplied by itself, the product is a square number y. Square roots are both negative and positive. For example, the square root of 25 is 5 and -5.

All non-negative real numbers m have a unique non-negative square root. This is referred to as the principal square root and is denoted by b, and is the radical sign.

For example, 5 is the principal square root of 25 as 5 * 5 = 5 2 = 25 and 5 is a non-negative integer. 25 is referred to as the radicand. Thus, the radicand is the number whose square root is being considered.

History of Square Roots

The Yale Babylonian Collection YBC 7289 clay tablet created between 1800 BC to 1600 BC contains the square root of two representations.

The Rhind Mathematical Papyrus in 1650 BC shows that Egyptian used an inverse proportion method to extract square roots. The Sulba Sutras in Ancient India, which was dated even before 800 – 500 BC, details several theoretical and applied aspects of squares and square roots.

Aryabhata in Aryabhatiya gives a method for finding the square root of numbers consisting of several digits. Several other cultures and mathematicians, such as the Greeks and Chinese, came up with their own methods to find square roots.

Eventually, all these excerpts were combined to come up with what we know as the modern version of finding square roots.

Properties of Square Roots

■ A perfect square number will have a perfect square root

■ An even perfect square will always have an even square root

■ An odd perfect square will always have an odd square root

■ A perfect square cannot be negative; therefore, the square root of a negative number is not defined, or it comes under complex numbers

■ Numbers that end with 1, 4, 5, 6, and 9 will always have a square root

■ Numbers that end with 2, 3, 7, and 8 will not have a perfect square root

■ A square root is possible only for numbers that end with an even number of zeros. Thus, numbers having an odd number of zeros at the end will not have a square root

■ We can multiply two square roots

■ If two same square roots are multiplied, they give a non-square root number

Cube Roots

Cube roots of a number are similar to square roots. When a number n is multiplied by itself three times to give the number itself, it is called a cube root. It can be represented as n 3 = m. Thus we can say the cube roots of a number m are those numbers n that satisfy the given equation. Cube roots can be negative and positive numbers. The cube roots of -125 are -5, -5, -5, as when we cube -5, we get -125.

Properties of Cube Roots

1. The cube roots of odd numbers will always be an odd number

2. The cube roots of even numbers will always be an even number

3. The cube roots of a negative integer will always be negative

Conclusion

Square roots and cube roots prove to be complicated topics, especially when it comes to solving questions. Thus, kids can turn to a platform like Cuemath to build a strong foundation of mathematical concepts. The certified math tutors create a holistic learning environment for students to provide them with the best quality of education possible.

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What are Square Roots and Cube Roots? - Space Coast Daily

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Suppose we have an expression given by z2 = y. We can say the square root of a number y is a number such that the above equation is satisfied. [...]

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Square Roots

The square root of a number is the number that, when squared (multiplied by itself), is equal to the given number. For example, the square root of 16, denoted 16411(5/3)9 1/2 or , is 4, because ² = 4×4 = 16. The square root of 121, denoted , is 11, because ² = 121. = 5/3, because ² = 25/9. = 9, because ² = 81. To take the square root of a fraction, take the square root of the numerator and the square root of the denominator. The square root of a number is always positive.

All perfect squares have square roots that are whole numbers. All fractions that have a perfect square in both numerator and denominator have square roots that are rational numbers. For example, = 9/7. All other positive numbers have squares that are non-terminating, non- repeating decimals, or irrational numbers. For example, = 1.41421356... and = 2.19503572....

Square Roots of Negative Numbers

Since a positive number multiplied by itself (a positive number) is always positive, and a negative number multiplied by itself (a negative number) is always positive, a number squared is always positive. Therefore, we cannot take the square root of a negative number.

Taking a square root is almost the inverse operation of taking a square. Squaring a positive number and then taking the square root of the result does not change the number: = = = | 6| = 6, and = | - 7| = 7. = 7. Thus, we conclude that squaring any number and then taking the square root of the result is equivalent to taking the absolute value of the given number. For example, = 6. However, squaring a negative number and then taking the square root of the result is equivalent to taking the opposite of the negative number:

Taking the square root first and then squaring the result yields a slightly different case. When we take the square root of a positive number and then square the result, the number does not change: ( ) ² = 11 ² = 121. However, we cannot take the square root of a negative number and then square the result, for the simple reason that it is impossible to take the square root of a negative number.

Cube Roots and Higher Order Roots

A cube root is a number that, when cubed, is equal to the given number. It is denoted with an exponent of "1/3". For example, the cube root of 27 is . The cube root of 125/343 is .

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Powers, Exponents, and Roots: Square Roots | SparkNotes

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The square root of a number is the number that, when squared (multiplied by itself), is equal to the given number. For example, the square root of 16, denoted 16411(5/3)9 1/2 or , is 4, because 2 = 4×4 = 16. The square root of 121, denoted, is 11, because 2 = 121.

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Comparing a square root to another number can be rough, unless you remember that squaring is opposite of taking the square root. Then things get much easier! See how it's done in this tutorial.

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How Do You Figure Out if a Square Root is Bigger than Another Number?

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Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

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To find square root or cube root of a number is not an easy task. When you're giving a time-bound exam like CAT, CMAT, CET, NMAT, etc. this can drain you of your precious time. This is a worse deal when finding square or cube root is only part of a bigger problem, like in Data Interpretation or Compound Interest problems in Quantitative Aptitude.
So if your Mental Mathematics is a little weak, let us learn how to quickly and easily find square root or cube root of a number. This trick is sure to save you at least 40 seconds of calculations per question. At first you will find it difficult but with practice, you will be able to find square root or cube root of any number. Then let us start.

103 ² = 10609

Step 1. Add the number to the ones digit:
103 + 3 = 106
Step 2. Square the ones digit number (if the result is a single digit put a 0 in front of it):
3 ² = 09
Step 3. Place the result from Step 2 next to the result from Step 1: 10609

Step 1. Subtract the number from 100: 100- 97 = 3

Step 2. Subtract the number (from Step 1) from original number : 97-3 =94

Step 3. Square the result from Step 1 (if the result is a single digit put a 0 in front of it): 3 ² = 09

Step 4. Place the result from Step 3 next to the result from Step 2: 9409

Step 1. Subtract the number from 50: 50-48=2

Step 2. Subtract the result (from Step 1) from 25: 25-2 =23

Step 3. Square the result from Step 1 if the result is a single digit put a 0 in front of it ) : 2 ² = 04

Step 4. Place the result from Step 3 next to the result from Step 2: 2304

Step 1. Add 25 to the ones digit: 25 + 3 = 28

Step 2. Square the ones digit number ( if the result is a single digit put a 0 in front of it ) : 3 ² = 09

Step 3. Place the result from Step 2 next to the result from Step 1 : 2809

First we need to remember cubes of 1 to 10 and unit digits of these cubes. The figure below shows the unit digits of cubes (on the right) of numbers from 1 to 10 (on the left).

1 = 1

2 = 8

3 = 7

4 = 4

5 = 5

6 = 6

7 = 3

8 = 2

9 = 9

10 = 0

Now with reference to above we can definitely say that:

Whenever unit digit of a number is 9, the unit digit of the cube of that number will also be 9. Similarly, if the unit digit of a number is 9, the unit digit of the cube root of that number will also be 9. Similarly, if unit digit of a number is 2, unit digit of the cube of that number will be 8 and vice versa if unit digit of a number is 8, unit digit of the cube root of that number will be 2. Similarly, it will be applied to unit digits of other numbers as well.

DERIVING CUBE ROOT FROM REMAINING DIGITS

Let's see this with the help of an example. Note that this method works only if the number given is a perfect cube.

Find the cube root of 474552.

Unit digit of 474552 is 2. So we can say that unit digit of its cube root will be 8.

Now we find cube root of 447552 by deriving from remaining digits.

Let us consider the remaining digits leaving the last 3 digits. i.e. 474.

Since 474 comes in between cubes of 7 and 8.

So the ten's digit of the cube root will definitely be 7

i.e. cube root of 474552 will be 78.

Let us take another example.

Find the cube root of 250047.

Since the unit digit of the number is 7, so unit digit in the cube root will be 3.

Now we will consider 250.

Since, 6 ³ < 250 < 7 ³, So tens digit will be 6

So we find cube root of the number to be 63.

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Easy Tricks to Find Square Roots and Cube Roots - CetKing

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To find square root or cube root of a number is not an easy task. When you're giving a time-bound exam like CAT, CMAT, CET, NMAT, etc. this can drain you of your precious time. This is a worse deal when finding square or cube root is only part of a bigger problem, like in ...

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Square Roots & Cube Roots | Generation Genius

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Calculator Use

Convert a number to and from scientific notation, e notation, engineering notation, standard form, and real numbers. Enter a number or a decimal number or scientific notation and the calculator converts to scientific notation, e notation, engineering notation, standard form and word form formats.

To enter a number in scientific notation use a carat ^ to indicate the powers of 10. You can also enter numbers in e notation. Examples: 3.45 x 10^5 or 3.45e5.

Order of magnitude will also be identified for the calculated standard form. The order of magnitude when written in standard form, is the nth power of 10. For example, 3.4 x 10^5 has an order of magnitude of 5 since 10 is raised to the 5th power.

How to Convert a Number to Scientific Notation

The proper format for scientific notation is a x 10^b where a is a number or decimal number such that the absolute value of a is greater than or equal to one and less than ten or, 1 ≤ | a| < 10. b is the power of 10 required so that the scientific notation is mathematically equivalent to the original number.

1. Move the decimal point in your number until there is only one non-zero digit to the left of the decimal point. The resulting decimal number is a.

2. Count how many places you moved the decimal point. This number is b.

3. If you moved the decimal to the left b is positive.
If you moved the decimal to the rightb is negative.
If you did not need to move the decimalb = 0.

4. Write your scientific notation number as a x 10^b and read it as "a times 10 to the power of b."

5. Remove trailing 0's only if they were originally to the left of the decimal point.

Example: Convert 357,096 to Scientific Notation

Move the decimal 5 places to the left to get 3.57096

a = 3.57096

We moved the decimal to the left so b is positive

b = 5

The number 357,096 converted to scientific notation is 3.57096 x 10^5

Example: Convert 0.005600 to Scientific Notation

Move the decimal 3 places to the right and remove leading zeros to get 5.600

a = 5.600

We moved the decimal to the right so b is negative

b = -3

The number 0.005600 converted to scientific notation is 5.600 x 10^-3

Note that we do not remove the trailing 0's because they were originally to the right of the decimal and are therefore significant figures.

E notation is basically the same as scientific notation except that the letter e is substituted for "x 10^".

Convert Scientific Notation to a Real Number

Multiply the decimal number by 10 raised to the power indicated.

3.456 x 10^4 = 3.456 x 10,000 = 34560

3.456 x 10^-4 = 3.456 x .0001 = 0.0003456

Additional Resources

See the Scientific Notation Calculator to add, subtract, multiply and divide numbers in scientific notation or E notation.

To round significant figures use the Significant Figures Calculator.

To see how standard form is similar to scientific notation visit Standard Form Calculator.

If you need a scientific calculator see our resources on scientific calculators.

Last updated: October 13, 2021

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Convert numbers to scientific notation. Calculator for conversion of numbers into scientific notation and e notation. Converts to proper scientific notation format.

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Square root and Cube roots are the most important topics in Maths, especially for class 8 students. To find the square root of any number, we need to find a number which when multiplied twice by itself gives the original number. Similarly, to find the cube root of any number we need to find a number which when multiplied three times by itself gives the original number.

Symbol: The square root is denoted by the symbol '√', whereas the cube root is denoted by '∛'.
Examples:
Square Root and Cube Root Table

Memorizing the squares and the square roots of the first few numbers are almost elementary and it can help you to solve problems much faster rather than having to work on it. Following is the square roots list and Cube root list of the first 15 natural numbers.

How to find Square Root and Cube Root

To find the square root of the number, we have to determine which number was squared to get the original number. For example, if we have to find the root of 16, then as we know, when we multiply 4 by 4, the result is 16. Hence, √16 = 4. Similarly, if we have to find the cube root of a number, say 64, then it is easy to determine that the cube of 4 gives 64. So the cube root of 64 is 4. But if the numbers are very large, then to find the roots, we have to use the prime factorisation method. Let us see some examples.

Solved Examples
Example 1: Find Square root of 256.
Solution: Given: The number is 256.

Prime factorisation of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2 × 2) 2 = 16 ²

Taking roots on both the sides, we get;

√256 = 16.

Hence, 16 is the answer.
Example 2: Find the cube root of 512.
Solution:

Given: The number is 512.

Prime factorisation of 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2) 3 = 8 ³

Taking the cube root on both the sides we get;

Hence, 8 is the answer.
Related Links:
Video Lessons on Square Roots

Basically, square explains the area of a square , which is equal to 'side x side'. But what if we have to find the side of the square. In such a case, we need to find the square root of the area of the square. In the same way, a cube explains the volume of the cube, which is equal to 'side x side x side'. Now, if we have to find the length of the edges of the cube, then we need to determine the cube root of the volume of a cube.

If subtraction is the opposite of addition and division is the reverse method of multiplication, then square root and cube root are the inverse processes of finding squares and cubes of numbers. This concept has been widely explained in Class 8 syllabus. To find the square root as well cube root for a given number, we can use the prime factorisation method, which we will discuss here later.

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Square root and Cube roots are the most important topics in Maths, especially for class 8 students. To find the square root of any number, we need to find a number which when multiplied twice by itself gives the original number.

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Calculator Use

Use this calculator to find the cube root of positive or negative numbers. Given a number x, the cube root of x is a number a such that .

If x is positive a will be positive. If x is negative a will be negative. The Cube Root Calculator is a specialized form of our common Radicals Calculator.

Example Cube Roots

The 3rd root of 64, or 64 radical 3, or the cube root of 64 is written as \( \sqrt[3]{64} = 4 \).

The 3rd root of -64, or -64 radical 3, or the cube root of -64 is written as \( \sqrt[3]{-64} = -4 \).

The cube root of 8 is written as \( \sqrt[3]{8} = 2 \).

The cube root of 10 is written as \( \sqrt[3]{10} = 2.154435 \).

What is a Cube Root as an Exponent?

The cube root of x is the same as x raised to the 1/3 power. This is written as \( \sqrt[3]{x} = x^{\frac{1}{3}} \).

What is a Cube Root of a Negative Number?

The common definition of the cube root of a negative number is:
=

First take the cube root of the number and then make it negative.

Example Cube Roots of Negative Numbers

The cube root of -27 is written as \( \sqrt[3]{-27} = -3 \).

The cube root of -8 is written as \( \sqrt[3]{-8} = -2 \).

The cube root of -64 is written as \( \sqrt[3]{-64} = -4 \).

Cube Roots Resulting in the Integers 1 Through 10

Cube root of 1 is 1

Cube root of 8 is 2

Cube root of 27 is 3

Cube root of 64 is 4

Cube root of 125 is 5

Cube root of 216 is 6

Cube root of 343 is 7

Cube root of 512 is 8

Cube root of 729 is 9

Cube root of 1000 is 10

To calculate fractional exponents use our calculator for Fractional Exponents.

References

Weisstein, Eric W. "Cube Root." FromMathWorld -- A Wolfram Web Resource. Cube Root

Math is Fun Cube Roots

Last updated: August 17, 2023

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Calculate the cube root of positive and negative numbers. 3rd root calculator. The cube root of x is a number y such that y³ = x. The cube root of a negative number is negative.

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Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:

Like this:

Or this:

It makes it easy to use big and small values.

OK, How Does it Work?

Example: 700

Why is 700 written as 7 × 10² in Scientific Notation ?

Both 700 and have the same value, just shown in different ways.

Example: 4,900,000,000

1,000,000,000 = 10 ⁹ ,
so 4,900,000,000 = in Scientific Notation

The number is written in two parts:

Just the digits, with the decimal point placed after the first digit, followed by

× 10 to a power that puts the decimal point where it should be
(i.e. it shows how many places to move the decimal point).

In this example, 5326.6 is written as 5.3266 × 10³,
because 5326.6 = 5.3266 × 1000 = 5.3266 × 10 ³

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

images/sci-notation-num.js

Other Ways of Writing It
3.1 × 10^8
We can use the ^ symbol (above the 6 on a keyboard), as it is easy to type.

3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

Calculators often use "E" or "e" like this:

Example: 3.12E4 is the same as 3.12 × 10⁴

How to Do it
To figure out the power of 10, think "how many places do I move the decimal point?"
Example: 0.0055 is written 5.5 × 10⁻³

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10⁻³

Example: 3.2 is written 3.2 × 10⁰

We didn't have to move the decimal point at all, so the power is

But it is now in Scientific Notation

Check!

After putting the number in Scientific Notation, just check that:

The "digits" part is between 1 and 10 (it can be 1, but never 10)

The "power" part shows exactly how many places to move the decimal point

Why Use It?

Because it makes it easier when dealing with very big or very small numbers, which are common in Scientific and Engineering work.

Example: it is easier to write (and read) 1.3 × 10⁻⁹ than 0.0000000013

It can also make calculations easier, as in this example:

Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.
What is its volume?
Let's first convert the three lengths into scientific notation:

Then multiply the digits together (ignoring the ×10s):

2.56 × 1.4 × 2.75 = 9.856

Last, multiply the ×10s:

10⁻⁶ × 10⁻⁷ × 10⁻⁴ = 10⁻¹⁷ (easier than it looks, just add −6, −4 and −7 together)

The result is 9.856×10⁻¹⁷ m³

It is used a lot in Science:

Example: Suns, Moons and Planets

The Sun has a Mass of 1.988 × 10 ³⁰ kg.

Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg
(and that number gives a false sense of many digits of accuracy.)

It can also save space! Here is what happens when you double on each square of a chess board:

Values are rounded off, so 53,6870,912 is shown as just 5×10 ⁸

That last value, shown as 9×10 ¹⁸ is actually 9,223,372,036,854,775,808

Engineering Notation

Engineering Notation is like Scientific Notation, except that we only use powers of ten that are multiples of 3 (such as 10 ³, 10⁻³, 10 ¹² etc).

Examples:

Notice that the "digits" part can now be between 1 and 1,000 (it can be 1, but never 1,000).

The advantage is that we can replace the ×10s with Metric Numbers. So we can use standard words (such as thousand or million), prefixes (such as kilo, mega) or the symbol (k, M, etc)

Example: 19,300 meters is written 19.3 × 10³ m, or 19.3 km

Example: 0.00012 seconds is written 120 × 10⁻⁶ s, or 120 microseconds

1340, 1341, 1342, 1344, 1345, 5077, 5078, 5079, 1343, 1346, 1347, 5080

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Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers: Like this: Or this: It makes it easy to use big and small values. Why is 700 written as 7 × 102 in Scientific Notation ? Both 700 and have the same value, just shown in different ways.