Indices

Indices, Laws of Indices and Radical Equations

1. Indices

Indices (or exponents) are a shorthand way to express repeated multiplication of the same number. For example, 2³ means 2 multiplied by itself three times (2 × 2 × 2).

Here are some basic concepts related to indices:

• Positive Indices: When the exponent is a positive integer, it indicates the number of times the base is multiplied by itself. For example, aⁿ means a × a × a × ... (n times).
• Zero Index: Any non-zero number raised to the power of zero is equal to 1. For example, a⁰ = 1 (for a ≠ 0).
• Negative Indices: A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, a^-n = 1/aⁿ.
• Fractional Indices: A fractional exponent represents both a root and a power. For example, a^1/n = n√a and a^m/n = (n√a)^m.

Examples of Indices:

1. 2³ = 2 × 2 × 2 = 8

2. 3⁰ = 1

3. 5^-2 = 1/5² = 1/25

4. 8^1/3 = 3√8 = 2

5. 4^1/2 = √4 = 2

6. 10² = 10 × 10 = 100

7. 6^-1 = 1/6

8. 9^1/2 = √9 = 3

9. 2⁴ = 2 × 2 × 2 × 2 = 16

10. 7^-3 = 1/7³ = 1/343

2. Laws of Indices

The laws of indices are a set of rules that help simplify expressions with exponents. They are:

• Product Law: a^m × aⁿ = a^m+n
• Quotient Law: a^m / aⁿ = a^m-n
• Power of a Power: (a^m)ⁿ = a^m×n
• Power of a Product: (a × b)ⁿ = aⁿ × bⁿ
• Power of a Quotient: (a/b)ⁿ = aⁿ / bⁿ

Examples of Laws of Indices:

1. a³ × a⁴ = a³⁺⁴ = a⁷

2. b⁵ / b² = b^5-2 = b³

3. (2³)² = 2^3×2 = 2⁶

4. (3 × 4)² = 3² × 4² = 9 × 16 = 144

5. (6/2)³ = 6³ / 2³ = 216 / 8 = 27

6. 5² × 5³ = 5²⁺³ = 5⁵

7. (x²)³ = x^2×3 = x⁶

8. y⁵ / y⁴ = y^5-4 = y¹ = y

9. (a × b)⁰ = a⁰ × b⁰ = 1 × 1 = 1

10. (x/2)⁴ = x⁴ / 2⁴ = x⁴ / 16

3. Radical Equations

A radical equation is an equation that contains a radical expression, which involves roots (such as square roots or cube roots). Solving radical equations involves isolating the radical and eliminating it.

Examples of radical equations:

• Square root equation: √(x) = 4
• Cube root equation: ∛(x) = 3

Examples of Radical Equations:

1. √(x) = 4 → x = 4² = 16

2. ∛(x) = 3 → x = 3³ = 27

3. √(x + 3) = 5 → x + 3 = 5² → x + 3 = 25 → x = 22

4. ∛(x - 2) = 4 → x - 2 = 4³ → x - 2 = 64 → x = 66

5. √(2x + 1) = 3 → 2x + 1 = 3² → 2x + 1 = 9 → 2x = 8 → x = 4

6. ∛(x + 4) = 5 → x + 4 = 5³ → x + 4 = 125 → x = 121

7. √(3x - 2) = 7 → 3x - 2 = 7² → 3x - 2 = 49 → 3x = 51 → x = 17

8. ∛(x + 1) = 2 → x + 1 = 2³ → x + 1 = 8 → x = 7

9. √(x + 5) = 6 → x + 5 = 6² → x + 5 = 36 → x = 31

10. ∛(x - 1) = 5 → x - 1 = 5³ → x - 1 = 125 → x = 126

Solutions to Radical Equations

1. Equation: √(x) = 4

Solution: Square both sides:

√(x) = 4 → x = 4² = 16

Answer: x = 16

2. Equation: ∛(x) = 3

Solution: Cube both sides:

∛(x) = 3 → x = 3³ = 27

Answer: x = 27

3. Equation: √(x + 3) = 5

Solution: Square both sides:

√(x + 3) = 5 → x + 3 = 5² = 25

Now, subtract 3 from both sides:

x = 25 - 3 = 22

Answer: x = 22

4. Equation: ∛(x - 2) = 4

Solution: Cube both sides:

∛(x - 2) = 4 → x - 2 = 4³ = 64

Now, add 2 to both sides:

x = 64 + 2 = 66

Answer: x = 66

5. Equation: √(2x + 1) = 3

Solution: Square both sides:

√(2x + 1) = 3 → 2x + 1 = 3² = 9

Now, subtract 1 from both sides:

2x = 9 - 1 = 8

Now, divide by 2:

x = 8 / 2 = 4

Answer: x = 4

6. Equation: ∛(x + 4) = 5

Solution: Cube both sides:

∛(x + 4) = 5 → x + 4 = 5³ = 125

Now, subtract 4 from both sides:

x = 125 - 4 = 121

Answer: x = 121

7. Equation: √(3x - 2) = 7

Solution: Square both sides:

√(3x - 2) = 7 → 3x - 2 = 7² = 49

Now, add 2 to both sides:

3x = 49 + 2 = 51

Now, divide by 3:

x = 51 / 3 = 17

Answer: x = 17

8. Equation: ∛(x + 1) = 2

Solution: Cube both sides:

∛(x + 1) = 2 → x + 1 = 2³ = 8

Now, subtract 1 from both sides:

x = 8 - 1 = 7

Answer: x = 7

9. Equation: √(x + 5) = 6

Solution: Square both sides:

√(x + 5) = 6 → x + 5 = 6² = 36

Now, subtract 5 from both sides:

x = 36 - 5 = 31

Answer: x = 31

10. Equation: ∛(x - 1) = 5

Solution: Cube both sides:

∛(x - 1) = 5 → x - 1 = 5³ = 125

Now, add 1 to both sides:

x = 125 + 1 = 126

Answer: x = 126