Understanding Powers of Ten in Mathematics

The power of ten in mathematics is defined as any of the integer powers of a number multiplied by ten. To put it another way, we add ten to itself, a specific number of times (when the power is a positive integer). Also, the number 1 is a power of ten (the zeroth power) in its definition. In this article, you will understand the power of ten, its facts, converting numbers in the power of ten, & Scientific Notation Regarding the Power of 10. So, let us start by understanding the power of ten in the coming section.

Power of Ten

In Mathematics, the power of 10 is any whole-valued (integer) power of the number 10. In other words, the power of 10 states that the 10 multiplied to itself n number of times (when the power is any positive integer). Hence, the 10 power in long-form is the number 1 followed by n zeroes where n is the number that is greater than 0. For example, 10⁷ is written as 1,00,00,000.  

When n is the number that is smaller than 0, the 10 power is found by multiplying the base value  10 ‘n’ times in the denominator and placing 1 in the numerator. For example, 10⁻³ is written as

\[\frac{1}{10*10*10}\] = 0.001

When n is equal to 0, the power of 10 is equal to 1. For example, 10⁰ = 1.

Read below to know the power of 10 Maths in detail.

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How to Convert Numbers in Power of 10 Maths?

To convert any number into the power of ten Math, two basic rules are followed.

1. If the number is given in the decimal notation, move the decimal point to the right side of its original position and place the decimal point after the first non zero digits. The power ten will be the number of places the original decimal point was moved and it will be negative as it was moved towards the right side.

Example:  0.0000732 = 7.32 x 10⁻⁵

2. If the whole number greater than 10 is to be changed into the power of 10 Math, then move the decimal point to the left side of its original position and place the decimal point after the first digit. The power of 10 will be the number of places the original decimal point was moved and it will be positive as it was moved towards the left side.

Example: 145,000 = 1.45 x 10⁵

Multiplying and Dividing By Positive Power of 10 Maths

1. When multiplying the number by the power of 10, we move the decimal points to the right side for each power of 10.

Example:

62.54 x 10¹ = 625.4

Here, the decimal point is shifted by one place to the right side.

62.54 x 10² = 6254.

Here, the decimal point is shifted by one place to the right side.

2. When dividing the number by the power of 10, we move the decimal points to the left side for each power of 10.

62.54 ÷ 10¹ = 6.254

Here, the decimal point is shifted one place to the left side.

62.54 ÷ 10² = 0. 6254.

Here, the decimal point is shifted two places to the left side.

Multiplying By the Negative Power of 10

Negative power tells how many times to divide the base number. When multiplying the number by the negative power of 10, we move the decimal points to the left side for each power of 10.

Example: 

6 x 10⁻³ = 6 x 1/10 x 1/10 x 1/10 = 6/1000 = 6 x 0.001 =  0.006

6.1 x 10⁻³ = 6.1 x 1/10 x 1/10 x 1/10 = 6.1/1000 = 6.1 x 0.001 = 0.0061

Scientific Notation Regarding Power of 10 Maths

The scientific notation, also known as the standard form, was given its name because it was first used by scientists to represent extremely small and large numbers. Exponents refer to the power of 10 multiplied by another number. Moreover, we can find them in both positive and negative forms.

Additionally, the positive form denotes multiplication, while the negative form represents division. The index of ten indicates how many places the decimal points should be moved to the right in the notation. 

In scientific notation, the numbers are represented in the form of a x 10ⁿ, where the variable a is the decimal with 1 ≤ a < 10ⁿ, and n is the integer.

To understand this, consider multiplying 1.35 by 10 to the fourth power. Alternatively, 1.35x10⁴.

You can then calculate it by 1.35 x  (10 x 10 x 10 x 10), or 1.35 x 10,000, to get the answer 13,500. Now, if we shift the decimal place to 1.35 over four places, we get 13,500.

Example:

Avogadro's number in scientific notation is approximately written as 6.022141793 x 10²³. Here a is the decimal 6.022141793 and n is the exponent 23. 

Facts to Remember

• A power 10 with a positive exponent such that 10⁴, means that the decimal point is shifted towards the left.

• A power 10 with a negative exponent such that 10⁴, means that the decimal point is shifted towards the right.

Solved Example

1. What is 2.35 x 10⁴?

Solution:

2.35 x 10⁴ can be calculated as 2.35 x (10 x 10 x 10 x 10) = 2.35 x 10000 

When multiplying the number by the power of 10, we move the decimal points to the right side for each power of 10

Accordingly, 

2.35 x 10000 = 2,35,000

2. How Do You Write 0.0002 in Scientific Notation?

Solution:

According to the rule, to convert 0.0002 in scientific notation,  we will move the decimal point to the right side of its original position and place the decimal point after the first non-zero digit. The power ten will be the number of places that will be negative as it was moved towards the right side.

Therefore, the scientific notation for 0.0002 is 2 x 10⁻⁴

3. Can You Help Sam to Write 9.56 x 10¹¹ in Standard Notation?

Solution:

Here 9.56 is 956. Now, Sam will move the decimal point 11 places to the right side and add trailing zeros accordingly. 

Therefore, the standard notation for  9.56 x 10¹¹ is 956,000,000, 000.

4. What is the notation form of 3,00,00,00,000 or 300 crores?

a. 3 ×10⁹

b. 3 × 10⁸

c. 3 × 10¹⁰

d. 3 × 10¹¹

Solution: The answer is option a- 3 ×10⁹  

Because, 3 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 3,00,00,00,000.

5. How to express 10 to the power of 10?

Solution: To find 10 to the power of 10, we can write it in the exponent form as 1010, where 10 is the base and 10 is the power as well.

It means 10 is multiplied 10 times. 

So, 1010 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000,000.

Hence, 10 to the power of 10 can be expressed as 1010 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000,000.

Conclusion

The idea of continuous and comprehensive growth is mighty simple, better learning. What helps in that, you may ask? Practising mathematical sums regularly helps remember facts and inculcate a better learning habit.