Power of Ten

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10 Power

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 \[\frac{1}{10\times10\times10}\] = 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 Math in detail.


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How to Convert Numbers in Power of 10 Math?

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 digit. 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 Math

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.

Example

The Avogadro's number in scientific notation is approximately is 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 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

FAQ (Frequently Asked Questions)

1. What is the Power of 10?

Ans. The power of 10 is the number of places the decimal point can be moved to express any number as a product of 10. The positive power 10 shows that the decimal point is moved to that number of places on the left side. The negative power 10 shows that the decimal point is moved to that number of places to the right side.

2. What Pattern Does the Power of Ten Maths Follow?

Ans. If you carefully observe then you will find that the number of trailing zeroes in the solutions is equal to the exponents. For example, if the power of 10 is 3 then the answer is 1000 which has 3 zeroes. Similarly, if the power of 10 increases to 6 then the answer 1,000,000 has 6 zeros. 

3. What Does Scientific Notation Mean?

Ans. Scientific notation is a way of representing very large or small numbers conveniently.  For example, For example, instead of writing 0.0000000046, we write 4.6 x 10-9. Scientific notation is widely used by Scientists, Engineers, and mathematicians because it simplifies certain arithmetic operations.