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Standard Form Formula in Exponents and Powers can be defined as expressing a number as a product of a base number and integral power of 10. The exponent of 10 will be positive for very large numbers and will be negative for very small numbers. The exponents of the numbers may be positive or negative. There are certain laws of exponents that are applicable for both positive and negative exponents.
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Key Terms: Negative Exponents, Laws of Exponents, Formulas for Exponent and Powers
Negative Exponents
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When the number is small, it can be expressed as a number to a negative power. When a base ‘x’ is raised to the power ‘-n’, it is denoted as x-n. It can also be expressed as (1/xn) where xn is in the denominator.
Example 1: 7-2
7-2 can be expressed as (1/72).
Laws of Exponents
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The laws of exponents defined for positive exponents hold true for negative exponents as well. The laws of exponents along with examples are tabulated below:
| Laws | Examples |
|---|---|
| a-m =1/am for any non-zero integer 'a' | 2-5= 1/25 |
| a0= 1 for any non-zero integer 'a' | 20 = 1 |
| a¹= a for any non-zero integer 'a' | 2¹ = 2 |
| am × an = am+n for any non-zero integer 'a' | 2³ × 24 = 2(3+4)= 27 2-4× 26 = 2(-4+6) = 2² |
| am / an =a(m-n) for any non-zero integer 'a' | 24/22 = 2(4-2) = 2² 2-5/23 = 2(-5-3) = 2-8 |
| (ab)m= am×bm for any non-zero integers 'a' and 'b' | (2×3)-3 = 2-3 × 3-3 |
| (a/b)m = am/ bm for any non-zero integers 'a' and 'b' | (2/3)-4 = 2-4 / 3-4 |
| (am)n = a(m x n) for any non- zero integer 'a' | (22)3 = 2(2 X 3) = 26 |
Standard Form Formula for Exponents and Powers
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In some situations, the number is very huge and it becomes difficult to understand the value of the number. Examples of such situations are as follows:
- Age of the Earth is said to be 4,600,000,000 years
- Surface area of the sun is said to be 6,090,000,000,000 sq. km.
- The distance between the sun and the Earth is said to be 149,600,000,000 m.
In some situations, the number is so small that it is difficult to read and understand in the decimal decimal form. Examples of such situations are as follows:
- The average diameter of a white blood cell is 14 - 16 micrometres or 0.00014 - 0.00016.
- The thickness of a paper is said to be 0.0016cm.
- The diameter of the wire that is connected inside a computer chip is 0.000003 m.
In both these situations expressing the numbers in an understandable manner is a challenge. To make the numbers readable, such numbers are expressed using the standard form formula.
The standard form formula for Exponents and Powers can be represented as follows:
a .b ×10n
Where a is any non-zero number from 1 to 9, b can be a single or multiple-digit number after the decimal, and n is the number of times 10 is repeated in the number. The value of n can be positive or negative depending on the value of the number. This standard form is also known as exponential notation or scientific notation.
The steps to convert a very large number to the standard form are as follows:
- Place the decimal point right after the first number from the left.
- Count the number of positions to the right of the decimal.
- Drop the zeros present at the end of the number and mention the number of places obtained in step 2 as a positive exponent to the base of 10.
- Now express the number as a product of the decimal number obtained in step 3 and the integral power of 10 obtained in step 3.
Example:
The distance between the sun and the Earth is 149,600,000,000 m.
This large number can be expressed in the standard form as 1.496 × 1011 m
Similarly to convert very small numbers to the standard form, the following steps should be followed:
- Place the decimal point immediately to the right of the first digit from the right.
- Count the number of positions to the left of the decimal.
- Drop the zeros present before the number and mention the number of places obtained in step 2 as a negative exponent to the base of 10.
- Now, express the number as a product of the decimal number obtained in step 3 and the integral power of 10 obtained in step 3.
Example:
The diameter of a wire connected inside a computer chip is 0.000003 m.
The number 0.000003 can be written as follows:
0.000003 = 3/1000000 = 3 × 10-6 m.
Therefore the number 0.000003 m can be expressed as 3 × 10-6 m.
Things to Remember
- When a number is small, it can be expressed using a negative exponent. The number can be written as x-n where x is the base and n is the exponent value.
- The laws of exponents are the same for numbers with positive exponents and negative exponents.
- The standard form formula is a.b × 10n where a is the digits on the left of the decimal, b is the digits on the right of the decimal and n is the exponent value which may be positive or negative depending on the value of the number.
- If the number is very large, it is expressed with a positive exponent value.
- If the number is very small, it is expressed with a negative exponent value.
Sample Questions
Ques. What is the standard form of 31860000000? (1 Marks)
Ans. The standard form of 31860000000 is 3.186 × 1010.
Ques. How do you write 0.00000356 in scientific notation? (1 Marks)
Ans. The scientific notation of 0.00000356 is 3.56 × 10-6.
Ques. What is the standard form in maths powers and exponents? (2 Marks)
Ans. When a number is expressed as a product of a number in the range of 1 to 9 and the integral exponent of the base number 10, then it is said to be in standard form.
Ques. What are the laws of exponents that are followed by negative exponents? (5 Marks)
Ans. The laws of exponents that are followed by negative exponents are as follows:
Ques. How do you write 72 crores in standard form? (1 Marks)
Ans. 72 crore can be written as 72,00,00,000 which can further be expressed in standard form as 7.2 × 108.
Ques. Write 0.0056 in standard form. (1 Marks)
Ans. 0.0056 = 56/10000 = (5.6 × 10)/10000 = 5.6 × 10 ×10?? = 5.6 × 10-3
Ques. What are exponents and powers? (2 Marks)
Ans. In general usage, the terms, exponents and powers are used interchangeably. In brief, exponents can be defined as the quantity obtained after raising a number to a certain power whereas the power of a number is defined as the repeated multiplication of a number a certain number of times.
Ques. What is the standard form of 0.000000000000874? (1 Marks)
Ans. The standard form of 0.000000000000874 is 8.74 × 10-13.
Ques. Express the number 6.53 × 10? in the usual form. (1 Marks)
Ans. 6.53 × 105 = 6.53 × 100000 = 653000
The usual form of the number is 653000.
Ques. How can the Standard Form Formula be represented? (2 Marks)
Ans. The Standard Form Formula can be represented as a.b × 10n where a is any non-zero number from 1 to 9, b can be a single or multiple-digit number after the decimal, and n is the number of times 10 is repeated in the number. The value of n can be positive or negative depending on the value of the number.
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