Grade 8 Indices and Cube Root

Concept: **Indices** (or exponents or powers) are a way to show repeated multiplication of the same number.

Notation: In the expression aⁿ:

• 'a' is the **base** (the number being multiplied).
• 'n' is the **index** (or exponent or power) (the number of times the base is multiplied by itself).

aⁿ = a * a * a * ... (n times)

Examples:

• 2³ = 2 * 2 * 2 = 8 (Base is 2, Index is 3)
• 5² = 5 * 5 = 25 (Base is 5, Index is 2)
• x⁴ = x * x * x * x (Base is x, Index is 4)

Law 1: Multiplication with the same base

When multiplying terms with the same base, add the indices.

a^m * aⁿ = a^{m + n}

Example: 2³ * 2⁴ = 2³⁺⁴ = 2⁷ = 128

Explanation: (2*2*2) * (2*2*2*2) = 2*2*2*2*2*2*2 (7 times)

Law 2: Division with the same base

When dividing terms with the same base, subtract the indices.

a^m / aⁿ = a^{m - n} (where a is not 0)

Example: 5⁶ / 5² = 5^6-2 = 5⁴ = 625

Explanation: (5*5*5*5*5*5) / (5*5) = 5*5*5*5 (4 times)

Law 3: Power of a Power

When raising a power to another power, multiply the indices.

(a^m)ⁿ = a^{m * n}

Example: (3²)³ = 3^2*3 = 3⁶ = 729

Explanation: (3*3) * (3*3) * (3*3) = 3*3*3*3*3*3 (6 times)

Law 4: Zero Index

Any non-zero number raised to the power of zero is equal to 1.

a⁰ = 1 (where a is not 0)

Example: 7⁰ = 1, (100)⁰ = 1, x⁰ = 1 (if x is not 0)

Explanation: Using the division law, a^m / a^m = a^m-m = a⁰. Also, a number divided by itself is 1 (a^m / a^m = 1). So, a⁰ must be equal to 1.

Law 5: Negative Index

A term with a negative index is equal to the reciprocal of the term with a positive index.

a^-n = 1 / aⁿ (where a is not 0)

1 / a^-n = aⁿ (where a is not 0)

Example: 2^-3 = 1 / 2³ = 1/8

Example: 1 / 5^-2 = 5² = 25

Explanation: Using the division law, a² / a⁵ = a^2-5 = a^-3. Also, a² / a⁵ = (a*a) / (a*a*a*a*a) = 1 / (a*a*a) = 1 / a³. So, a^-3 must be equal to 1 / a³.

Cube: The result of multiplying a number by itself three times.

Example: The cube of 2 is 2 * 2 * 2 = 8. (2³ = 8)

Example: The cube of 4 is 4 * 4 * 4 = 64. (4³ = 64)

Perfect Cube: A number that is the cube of an integer.

Example: 1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³) are perfect cubes.

Cube Root: The number that, when multiplied by itself three times, gives the original number. It is the inverse operation of cubing a number.

Notation: The symbol for cube root is ³√.

³√x means the cube root of x.

Example: ³√8 = 2 (because 2 * 2 * 2 = 8)

Example: ³√64 = 4 (because 4 * 4 * 4 = 64)

Step 1: Find the prime factorization of the number (216).

2 | 216
2 | 108
2 |  54
3 |  27
3 |   9
3 |   3
  |   1
                                    

Prime factorization of 216 = 2 * 2 * 2 * 3 * 3 * 3

Step 2: Group the prime factors in triplets (groups of three identical factors).

216 = (2 * 2 * 2) * (3 * 3 * 3)

Step 3: Take one factor from each triplet and multiply them together.

Take one '2' from the first triplet and one '3' from the second triplet.

Cube root of 216 = 2 * 3

Step 4: Calculate the product.

2 * 3 = 6

Result: The cube root of 216 is 6.

³√216 = 6