Indices - Laws of Exponents
Key Concepts
Solved Examples: 8
Practice Questions: 3
Solved Examples
Question 1: Find the value of: 5³ × 5⁰ + 2⁴
We know: 5⁰ = 1
Therefore,
5³ × 5⁰ + 2⁴
= 5³ × 1 + 16
= 125 + 16
= 141
Therefore,
5³ × 5⁰ + 2⁴
= 5³ × 1 + 16
= 125 + 16
= 141
Question 2: Find the value of (3⁴ × 9²) ÷ 27²
Convert all terms into the same base:
9 = 3²
27 = 3³
Therefore,
(3⁴ × (3²)²) ÷ (3³)²
= (3⁴ × 3⁴) ÷ 3⁶
Using laws of indices:
= 3⁸ ÷ 3⁶
= 3⁸⁻⁶
= 3²
= 9
9 = 3²
27 = 3³
Therefore,
(3⁴ × (3²)²) ÷ (3³)²
= (3⁴ × 3⁴) ÷ 3⁶
Using laws of indices:
= 3⁸ ÷ 3⁶
= 3⁸⁻⁶
= 3²
= 9
Question 3: 31⁷·⁵ ÷ 31³⁄² × 31⁻³ = (√31)ˣ
31⁷·⁵ ÷ 31³⁄² × 31⁻³ = (√31)ˣ
or, 31⁷·⁵ ÷ 31¹·⁵ × 31⁻³ = (√31)ˣ
or, 31⁷·⁵ ⁻ ¹·⁵ ⁻ ³ = (√31)ˣ
or, 31³ = (√31)ˣ
or, 31³ = 31ˣ⁄²
Since the bases are equal
x/2 = 3,
or x = 6
or, 31⁷·⁵ ÷ 31¹·⁵ × 31⁻³ = (√31)ˣ
or, 31⁷·⁵ ⁻ ¹·⁵ ⁻ ³ = (√31)ˣ
or, 31³ = (√31)ˣ
or, 31³ = 31ˣ⁄²
Since the bases are equal
x/2 = 3,
or x = 6
Question 4: In the equation: (A/21) × (A/189) = 1 find the value of A.
Given, (A/21) × (A/189) = 1
Multiplying the fractions:
A²/(21 × 189) = 1
A²/3969 = 1
Therefore,
A² = 3969
A² = 63²
A = 63
Multiplying the fractions:
A²/(21 × 189) = 1
A²/3969 = 1
Therefore,
A² = 3969
A² = 63²
A = 63
Question 5: Which of the following is larger 2³⁰⁰ or 3²⁰⁰?
Rewrite both expressions with the same power:
2³⁰⁰ = (2³)¹⁰⁰
3²⁰⁰ = (3²)¹⁰⁰
Now compare the bases:
2³ = 8
3² = 9
Therefore,
(2³)¹⁰⁰ = 8¹⁰⁰
(3²)¹⁰⁰ = 9¹⁰⁰
Since, 9¹⁰⁰ > 8¹⁰⁰
Therefore, 3²⁰⁰ > 2³⁰⁰
2³⁰⁰ = (2³)¹⁰⁰
3²⁰⁰ = (3²)¹⁰⁰
Now compare the bases:
2³ = 8
3² = 9
Therefore,
(2³)¹⁰⁰ = 8¹⁰⁰
(3²)¹⁰⁰ = 9¹⁰⁰
Since, 9¹⁰⁰ > 8¹⁰⁰
Therefore, 3²⁰⁰ > 2³⁰⁰
Question 6: Find the value of x from the equation: 2ˣ × 8¹⁄⁴ = 2¹⁄⁴
Convert 8 into a power of 2: 8 = 2³
Therefore,
8¹⁄⁴ = (2³)¹⁄⁴
= 2³⁄⁴
Now the equation becomes:
2ˣ × 2³⁄⁴ = 2¹⁄⁴
Therefore,
2ˣ⁺³⁄⁴ = 2¹⁄⁴
Since the bases are the same:
x + 3⁄4 = 1⁄4
x = 1⁄4 − 3⁄4
x = −2⁄4
x = −1⁄2
Therefore,
8¹⁄⁴ = (2³)¹⁄⁴
= 2³⁄⁴
Now the equation becomes:
2ˣ × 2³⁄⁴ = 2¹⁄⁴
Therefore,
2ˣ⁺³⁄⁴ = 2¹⁄⁴
Since the bases are the same:
x + 3⁄4 = 1⁄4
x = 1⁄4 − 3⁄4
x = −2⁄4
x = −1⁄2
Question 7: If 9ˣ − 9ˣ⁻¹ = 648 find the value of xˣ
Given,
9ˣ − 9ˣ⁻¹ = 648
Taking 9ˣ⁻¹ common:
9ˣ⁻¹(9 − 1) = 648
9ˣ⁻¹ × 8 = 648
9ˣ⁻¹ = 648 ÷ 8
9ˣ⁻¹ = 81
Since,
81 = 9²
Therefore,
9ˣ⁻¹ = 9²
Comparing the powers:
x − 1 = 2
x = 3
Now,
xˣ = 3³
= 27
9ˣ − 9ˣ⁻¹ = 648
Taking 9ˣ⁻¹ common:
9ˣ⁻¹(9 − 1) = 648
9ˣ⁻¹ × 8 = 648
9ˣ⁻¹ = 648 ÷ 8
9ˣ⁻¹ = 81
Since,
81 = 9²
Therefore,
9ˣ⁻¹ = 9²
Comparing the powers:
x − 1 = 2
x = 3
Now,
xˣ = 3³
= 27
Question 8: If 4ˣ⁻ʸ = 64 and 4ˣ⁺ʸ = 1024 find the value of x.
Convert the numbers into powers of 4:
64 = 4³
1024 = 4⁵
Therefore, 4ˣ⁻ʸ = 4³
Comparing the powers: x − y = 3 … (1)
Also,
4ˣ⁺ʸ = 4⁵
Comparing the powers: x + y = 5 … (2)
Adding equations (1) and (2):
(x − y) + (x + y) = 3 + 5
2x = 8
x = 4
64 = 4³
1024 = 4⁵
Therefore, 4ˣ⁻ʸ = 4³
Comparing the powers: x − y = 3 … (1)
Also,
4ˣ⁺ʸ = 4⁵
Comparing the powers: x + y = 5 … (2)
Adding equations (1) and (2):
(x − y) + (x + y) = 3 + 5
2x = 8
x = 4
Question 9: If 2²ⁿ⁻¹ = 1/8ⁿ⁻³ find the value of n.
Given: 2²ⁿ⁻¹ = 1/8ⁿ⁻³
Convert 8 into a power of 2: 8 = 2³
Therefore,
1/8ⁿ⁻³ = 1/(2³)ⁿ⁻³
= 1/2³ⁿ⁻⁹
= 2⁻³ⁿ⁺⁹
Now the equation becomes:
2²ⁿ⁻¹ = 2⁻³ⁿ⁺⁹
Since the bases are the same, equate the powers:
2n − 1 = −3n + 9
5n = 10
n = 2
Convert 8 into a power of 2: 8 = 2³
Therefore,
1/8ⁿ⁻³ = 1/(2³)ⁿ⁻³
= 1/2³ⁿ⁻⁹
= 2⁻³ⁿ⁺⁹
Now the equation becomes:
2²ⁿ⁻¹ = 2⁻³ⁿ⁺⁹
Since the bases are the same, equate the powers:
2n − 1 = −3n + 9
5n = 10
n = 2
Question 10: If 2ⁿ⁺⁴ − 2ⁿ⁺² = 3 find the value of n.
Given: 2ⁿ⁺⁴ − 2ⁿ⁺² = 3
Taking 2ⁿ⁺² common:
2ⁿ⁺²(2² − 1) = 3
2ⁿ⁺²(4 − 1) = 3
3 × 2ⁿ⁺² = 3
Dividing both sides by 3:
2ⁿ⁺² = 1
Since, 2⁰ = 1
Therefore,
n + 2 = 0
n = −2
Taking 2ⁿ⁺² common:
2ⁿ⁺²(2² − 1) = 3
2ⁿ⁺²(4 − 1) = 3
3 × 2ⁿ⁺² = 3
Dividing both sides by 3:
2ⁿ⁺² = 1
Since, 2⁰ = 1
Therefore,
n + 2 = 0
n = −2
Question: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
For divisibility by 11, the difference between the sum of the digits in alternate positions must be 0 or a multiple of 11.
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Question: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
For divisibility by 11, the difference between the sum of the digits in alternate positions must be 0 or a multiple of 11.
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Question: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
For divisibility by 11, the difference between the sum of the digits in alternate positions must be 0 or a multiple of 11.
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Question: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
For divisibility by 11, the difference between the sum of the digits in alternate positions must be 0 or a multiple of 11.
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Question: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
For divisibility by 11, the difference between the sum of the digits in alternate positions must be 0 or a multiple of 11.
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Question: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
For divisibility by 11, the difference between the sum of the digits in alternate positions must be 0 or a multiple of 11.
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
What are Indices?
An index (plural: indices) is a small number written slightly above and to the right of another number or variable. It indicates how many times the base is multiplied by itself. Indices provide a compact way to represent repeated multiplication and simplify lengthy mathematical expressions.
For example, 5³ means 5 × 5 × 5, where 5 is the base and 3 is the index (or exponent). Similarly, x⁴ means x × x × x × x.
Parts of an Exponential Expression
Every exponential expression consists of two parts: the base and the index (or exponent). The base is the number or variable that is multiplied repeatedly, while the index indicates how many times the base is multiplied by itself.
Examples:
• In 7⁴, the base is 7 and the index is 4.
• In x⁵, the base is x and the index is 5.
• In (ab)³, the base is ab and the index is 3.
Examples:
• In 7⁴, the base is 7 and the index is 4.
• In x⁵, the base is x and the index is 5.
• In (ab)³, the base is ab and the index is 3.
Understanding Repeated Multiplication
An index represents repeated multiplication of the base by itself. Instead of writing the same factor multiple times, indices provide a shorter and more convenient form.
Examples:
2⁵ = 2 × 2 × 2 × 2 × 2 = 32
4³ = 4 × 4 × 4 = 64
x⁴ = x × x × x × x
Examples:
2⁵ = 2 × 2 × 2 × 2 × 2 = 32
4³ = 4 × 4 × 4 = 64
x⁴ = x × x × x × x
Types of Indices
Indices can be positive, zero or negative. Each type follows a specific rule.
Positive Index: A positive index indicates repeated multiplication.
Example: 3⁴ = 3 × 3 × 3 × 3 = 81
Zero Index: Any non-zero number raised to the power 0 is equal to 1.
Examples: 8⁰ = 1, x⁰ = 1, where x ≠ 0
Negative Index: A negative index represents the reciprocal of the corresponding positive index.
Examples: 2⁻³ = 1/2³ = 1/8, x⁻² = 1/x²
Positive Index: A positive index indicates repeated multiplication.
Example: 3⁴ = 3 × 3 × 3 × 3 = 81
Zero Index: Any non-zero number raised to the power 0 is equal to 1.
Examples: 8⁰ = 1, x⁰ = 1, where x ≠ 0
Negative Index: A negative index represents the reciprocal of the corresponding positive index.
Examples: 2⁻³ = 1/2³ = 1/8, x⁻² = 1/x²
Laws of Exponents
A number is divisible by 5 if its last digit is 0 or 5.
Examples: 85, 320, and 1,745 are divisible by 5 because they end in 0 or 5.
Examples: 85, 320, and 1,745 are divisible by 5 because they end in 0 or 5.
Divisibility Rule for 6
Indices follow a set of rules known as the laws of exponents, which simplify calculations involving powers.
The important laws are:
• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ, where a ≠ 0
• (aᵐ)ⁿ = aᵐⁿ
• (ab)ⁿ = aⁿbⁿ
• (a/b)ⁿ = aⁿ/bⁿ, where b ≠ 0
• a⁰ = 1, where a ≠ 0
• a⁻ⁿ = 1/aⁿ, where a ≠ 0
These laws are the foundation for simplifying exponential expressions.
The important laws are:
• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ, where a ≠ 0
• (aᵐ)ⁿ = aᵐⁿ
• (ab)ⁿ = aⁿbⁿ
• (a/b)ⁿ = aⁿ/bⁿ, where b ≠ 0
• a⁰ = 1, where a ≠ 0
• a⁻ⁿ = 1/aⁿ, where a ≠ 0
These laws are the foundation for simplifying exponential expressions.
Summary of Indices - Laws of Exponents
| Concept | Summary |
|---|---|
| Index (Exponent) | A small number written above and to the right of a base that indicates repeated multiplication. |
| Base | The number or variable that is repeatedly multiplied. |
| Positive Index | Represents repeated multiplication of the base. |
| Zero Index | For any non-zero base, a⁰ = 1. |
| Negative Index | Represents the reciprocal of the corresponding positive index. a⁻ⁿ = 1/aⁿ. |
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0) |
| Power of a Power | (aᵐ)ⁿ = aᵐⁿ |
| Power of a Product | (ab)ⁿ = aⁿbⁿ |
| Power of a Quotient | (a/b)ⁿ = aⁿ/bⁿ (b ≠ 0) |
Common Mistakes
- Students often check the entire number instead of only the required digits. Example: For divisibility by 4, check only the last two digits, not the whole number.
- Confusing the rules for 3 and 9. Some students add the digits correctly but forget that
- The sum must be divisible by 3 for divisibility by 3.
- The sum must be divisible by 9 for divisibility by 9.
- Forgetting the rule for 6. A number is divisible by 6 only if it is divisible by both 2 and 3. Being divisible by just one of them is not enough.
- Using the wrong digits for 8. For divisibility by 8, check only the last three digits. Checking the last two digits gives the wrong result.
- Applying the rule for 11 incorrectly. When testing divisibility by 11, add the digits in alternate positions and find the difference between the two sums. Do not simply add all the digits together.
Practice Questions
Question: Find the greatest possible value of (a + b) for which the 8-digit number 143b203a is divisible by 15.
Question: If the number 59a44b is divisible by 36 then what is the maximum value of (a + b)?
Question: If a 5-digit number 535ab is divisible by 3, 7 and 11, then what is the value of (a – b)?
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