Laws of Indices and How to Use Powers of 10 in Calculations
In Mathematics, Index (indices) is the power or exponent which is raised to a number or a variable. For example, in number 25, 5 is the index of 2. The plural form of index is indices. In algebra, we come across constants and variables. The constant value is fixed that cannot be changed, whereas a variable can be assigned any number or we can say its value can be changed. In algebra, mostly we deal with indices in terms of numbers.
Index Definition
Index of a variable is a value that is raised to the power of the variable. The index value is also known as powers or exponents. It shows the number of times the given number has to be multiplied. It is represented in the form:
am = a × a × a × a .... × a(m times)
Here, a is called base and m is the index.
The index says that a particular number (or base) is to be multiplied by itself or the number of times equal to the index raised to it. It is an easy way of writing big numbers and calculations.
Laws of Indices
There are some fundamental rules or laws of indices that are necessary to understand before we start solving problems with indices. We follow these rules while performing algebraic operations on indices. Following are the rules we need to follow:
Rule 1: If a constant or variable has index as ‘0’, then the value of the result will be one. The base can be of any value.
a0 = 1
Example: 50 = 1,120 = 1
Rule 2: If the index value is negative, then it can be shown as the reciprocal of the positive index raised to the same variable.
a-p = 1/ap
Example: 5-1 = 1/5
Rule 3: If we multiply two variables with the same base, we need to add their powers and raise them to that base.
ap.aq = ap+q
Example: 52.53 = 52+3 = 55
Rule 4: To divide two variables with the same base, we need to subtract the power of the denominator from the power of the numerator and raise it to that base.
ap/aq = ap-q
Example:104/103 = 101
Rule 5: When a variable with some index is again raised with a different index, then both the indices are multiplied together raised to the power of the same base.
(ap)q = apq
Example: (82)4 = 88
Rule 6: When two variables with different bases, but the same indices are multiplied together, we have to multiply its base and raise the same index to multiplied variables.
ap.bp = (ab)p
Example:32.42 = (3.4)2 = 122
Rule 7: When two variables with different bases are given, and the same indices are divided. We have to divide the bases and raise the same index to it.
ap/bp = (a/b)p
Rule 8: If an index is in the form of a fraction it can be represented in the radical form.
\[a^{\frac{p}{q}} = \sqrt[q]{a^{p}}\]
Example: \[5^{\frac{1}{2}} = \sqrt{5}\]
2 to the power of 10 = 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024.
Powers of 10
"Powers of 10" is a very useful tool when we have to express large or small numbers.
Instead of having lots of zeros, we can show how many powers of 10 will make that many zeros.
Example: 5000 = 5 × 1000 = 5 × 103
5 thousand is 5 times a thousand. And a thousand is 103. So 5 times 103 = 5,000
10 to the power of 3 = 10 × 10 × 10 = 1000
In words, we can say 103 could be called "10 to the third power", "10 to the power 3" or simply "10 cubed".
10 to the power of 4 equals 10 × 10 × 10 × 10 = 10000.
In words, we can say 104 could be called "10 to the fourth power", "10 to the power 4" or "10 to the 4"
10 to the power of 6 = 10 × 10 × 10 × 10 × 10 × 10 = 1000000
Negative Powers of 10
A negative power means how many times we have to divide by the number.
Example: 10 ÷ 103 = 0.01
Other Way of Writing × symbol
Sometimes people use the ^ symbol instead of x, as it is easy to type.
Example: 3 × 10^4 is same as the 3 × 104
3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000
Note: For negative powers of 10: For negative powers of 10, move the decimal point to the left.
Solved Examples:
1. Find the value of \[27^{\frac{2}{3}}\]
Sol: Here index is in the form of a fraction. So we need to change it to radical form and then solve
\[27^{\frac{2}{3}} = \sqrt[3]{27^{2}}\]
27 can be written as 33
= \[\sqrt[3]{3^{6}}\]
= \[3^{\frac{6}{3}}\]
= 32
= 9
Hence the value of \[27^{\frac{2}{3}}\] is 9.
2. Express the value of 25 ÷ 22 in index notation.
Sol: According to the rule when two numbers with the same base are divided, we need to subtract the power of the denominator.
25 ÷ 22 = 25-2 = 23 = 8
Hence, the value of 25 ÷ 22 in index notation is 23.
Conclusion:
The index of 10 means many places we have to move the decimal point. A positive value means to move it to the right, whereas negative means to the left. An example is given below:
FAQs on Understanding Index Notation and Powers of 10
1. What is index notation in Maths?
Index notation is a way of writing repeated multiplication of the same number using powers or exponents.
- It is written as an, where:
- a = base number
- n = index or exponent
- Example: 53 = 5 × 5 × 5 = 125
Index notation makes calculations shorter and is widely used in algebra and powers of 10.
2. What are the basic laws of indices?
The basic laws of indices are rules used to simplify expressions involving powers.
- am × an = am+n
- am ÷ an = am−n
- (am)n = amn
- a0 = 1 (for a ≠ 0)
- a−n = 1 / an
These index laws are essential for simplifying algebraic expressions and powers of 10.
3. What is a power of 10?
A power of 10 is a number written as 10n, where n is an integer.
- 101 = 10
- 102 = 100
- 103 = 1000
Each increase in the exponent moves the decimal point one place to the right, which is useful in place value and scientific notation.
4. How do negative powers of 10 work?
A negative power of 10 means you divide by 10 repeatedly.
- 10−1 = 1/10 = 0.1
- 10−2 = 1/100 = 0.01
- 10−3 = 1/1000 = 0.001
Each negative exponent moves the decimal point one place to the left, which is important in decimals and scientific notation.
5. What is the difference between base and exponent?
The base is the number being multiplied, and the exponent shows how many times it is multiplied by itself.
- In 43:
- 4 is the base
- 3 is the exponent (index)
- It means 4 × 4 × 4 = 64
Understanding base and exponent is key to mastering index notation.
6. How do you multiply powers of 10?
To multiply powers of 10, add the exponents.
- Use the rule: 10a × 10b = 10a+b
- Example: 103 × 102 = 105
- 105 = 100,000
This rule comes from the law of indices for multiplication.
7. How do you divide powers of 10?
To divide powers of 10, subtract the exponents.
- Use the rule: 10a ÷ 10b = 10a−b
- Example: 106 ÷ 102 = 104
- 104 = 10,000
This makes calculations with large numbers faster and easier.
8. What is scientific notation and how is it related to powers of 10?
Scientific notation expresses numbers in the form a × 10n where 1 ≤ a < 10.
- Example: 4,500 = 4.5 × 103
- Example: 0.006 = 6 × 10−3
It uses powers of 10 to write very large or very small numbers in a compact form.
9. What does any number to the power of zero equal?
Any non-zero number raised to the power of zero equals 1.
- Rule: a0 = 1 (where a ≠ 0)
- Example: 70 = 1
- Example: 100 = 1
This rule comes from the division law of indices.
10. What are common mistakes when working with index notation and powers of 10?
Common mistakes include adding exponents incorrectly and misunderstanding negative powers.
- Adding exponents when bases are different (incorrect)
- Forgetting that a−n = 1 / an
- Thinking 10−2 is negative (it equals 0.01)
- Forgetting that a0 = 1
Carefully applying the laws of indices helps avoid these common errors.
