Learn about indices and their laws through these examples with detailed step-by-step explanations.
Example 1: What are Indices?
Explain the concept of indices (exponents).
Concept: Indices, also called exponents or powers, are a way to show repeated multiplication of the same number.
Notation: In the expression am:
'a' is called the **base**.
'm' is called the **index** or **exponent** or **power**.
Meaning: am means 'a' is multiplied by itself 'm' times.
Example: 53 means 5 x 5 x 5.
Calculation: 53 = 5 x 5 x 5 = 25 x 5 = 125.
Example 2: Law 1: Multiplication (Same Base)
Explain and apply the law: am x an = am+n.
Law: When multiplying numbers with the same base, add the exponents.
Formula: am x an = am+n
Example: Calculate 23 x 24.
Step 1: Identify the base and exponents. Base is 2, exponents are 3 and 4. The bases are the same.
Step 2: Apply the law: Keep the base and add the exponents.
23 x 24 = 23+4 = 27.
Step 3: Calculate the result.
27 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128.
Result: 23 x 24 = 128.
Example 3: Law 2: Division (Same Base)
Explain and apply the law: am ÷ an = am-n (where a is not 0).
Law: When dividing numbers with the same base, subtract the exponents.
Formula: am ÷ an = am-n (a ≠ 0)
Example: Calculate 56 ÷ 52.
Step 1: Identify the base and exponents. Base is 5, exponents are 6 and 2. The bases are the same.
Step 2: Apply the law: Keep the base and subtract the exponents (exponent of numerator minus exponent of denominator).
56 ÷ 52 = 56-2 = 54.
Step 3: Calculate the result.
54 = 5 x 5 x 5 x 5 = 625.
Result: 56 ÷ 52 = 625.
Example 4: Law 3: Power of a Power
Explain and apply the law: (am)n = am x n.
Law: When raising a power to another power, multiply the exponents.
Formula: (am)n = am x n
Example: Calculate (32)3.
Step 1: Identify the base and exponents. Base is 3, the inner exponent is 2, and the outer exponent is 3.
Step 2: Apply the law: Keep the base and multiply the exponents.
(32)3 = 32 x 3 = 36.
Step 3: Calculate the result.
36 = 3 x 3 x 3 x 3 x 3 x 3 = 729.
Result: (32)3 = 729.
Example 5: Law 4: Zero Index
Explain and apply the law: a0 = 1 (where a is not 0).
Law: Any non-zero number raised to the power of zero is equal to 1.
Formula: a0 = 1 (a ≠ 0)
Example: Calculate 70 and (-4)0.
Step 1: Identify the base and exponent. In both cases, the exponent is 0. The bases (7 and -4) are not zero.
Step 2: Apply the law: Any non-zero base to the power of 0 is 1.
70 = 1.
(-4)0 = 1.
Result: 70 = 1 and (-4)0 = 1.
Example 6: Law 5: Negative Index
Explain and apply the law: a-m = 1 / am (where a is not 0).
Law: A number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent.
Formula: a-m = 1 / am (a ≠ 0)
Example: Calculate 2-3.
Step 1: Identify the base and exponent. Base is 2, exponent is -3.
Step 2: Apply the law: Write the reciprocal of the base raised to the positive exponent.
2-3 = 1 / 23.
Step 3: Calculate the value of the positive power in the denominator.
23 = 2 x 2 x 2 = 8.
Step 4: Substitute the value back into the fraction.
1 / 23 = 1 / 8.
Result: 2-3 = 1/8.
Practice Mode
Enter a simple expression involving indices.
Note: This basic solver can evaluate expressions like "2^3", "5 to the power of -2", "calculate (1/2)^-1", "3^0 * 4^2", "10^5 / 10^3", "(2^3)^2". Use ^ or "to the power of" for exponents, * for multiplication, / for division, and parentheses for grouping.
Related Concepts
Explore these related mathematical concepts to deepen your understanding of indices.
Base
The number being multiplied repeatedly.
Exponent / Index / Power
The number that indicates how many times the base is multiplied by itself.
Repeated Multiplication
Multiplying a number by itself multiple times.
Reciprocal
Flipping a fraction (used with negative exponents).