How to Apply Index Notation and Powers of 10 in Maths Problems
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 Index Notation and Powers of 10 Made Easy
1. What is index notation in mathematics?
Index notation is a concise way of writing repeated multiplication of the same number. It consists of two parts: a base (the number being multiplied) and an exponent or index (which shows how many times the base is multiplied by itself). For example, instead of writing 5 x 5 x 5, we can use index notation to write it as 5³, where 5 is the base and 3 is the exponent.
2. How are powers of 10 represented using index notation?
Powers of 10 are represented by using 10 as the base. The exponent indicates the number of zeros after the 1 in the expanded number. For example:
- 100 = 10 x 10 = 10²
- 1,000 = 10 x 10 x 10 = 10³
- 10,000,000 = 10⁷
3. What do the special exponents 1, 0, and -1 mean for powers of 10?
The values of these special exponents follow specific rules that are fundamental in mathematics:
- Exponent of 1: Any number to the power of 1 is the number itself. So, 10¹ = 10.
- Exponent of 0: Any non-zero number to the power of 0 is always 1. So, 10⁰ = 1.
- Exponent of -1: A negative exponent indicates a reciprocal. So, 10⁻¹ is the same as 1/10 or 0.1.
4. How can we express a large number like 4,52,000 using powers of 10?
To express a large number like 4,52,000 using powers of 10, we often use scientific notation. You can write it as 4.52 x 100,000. Since 100,000 is 10⁵ (10 multiplied by itself 5 times), the number can be written concisely as 4.52 x 10⁵. This method is crucial for writing large numbers in a standard, manageable format.
5. Why is using index notation with powers of 10 so important in science?
Index notation with powers of 10 is essential in science because it simplifies calculations and the representation of extremely large or small quantities. Scientists frequently work with numbers like the distance to a star or the size of an atom. Writing these numbers out in full would be cumbersome and prone to errors. For example, the distance from Earth to the Sun, approximately 150,000,000 km, is much easier to write and use in equations as 1.5 x 10⁸ km.
6. How does a negative exponent for a power of 10 relate to decimal places?
A negative exponent with a base of 10 directly corresponds to the position of a digit after the decimal point. A negative exponent signifies division. For example, 10⁻² means 1 ÷ 10², which is 1/100 or 0.01. The exponent value (-2) indicates that the '1' is in the second decimal place (the hundredths place). Similarly, 10⁻³ is 0.001, placing the '1' in the third decimal place.
7. What is the key difference between a digit's face value and its place value when using powers of 10?
The face value of a digit is simply the digit itself, regardless of its position. In the number 782, the face value of '7' is just 7. However, the place value depends on its position and is best understood using powers of 10. The place value of '7' in 782 is 700, which can be expressed as 7 x 10². Index notation helps explain that a digit's true value in a number is its face value multiplied by a specific power of 10.
