Indices (Exponents) Worksheet for 15-Year-Olds — High School Math Practice & Laws of Indices

Instructions

This worksheet will test your understanding of the laws of indices (also known as exponents or powers). Read through the rules below and then answer the questions in each section. Simplify your answers fully. Give answers with positive indices where appropriate.

Section 1: Basic Simplification

Use the multiplication, division, and power of a power rules to simplify these expressions.

1. x⁵ × x⁴
2. y¹⁰ ÷ y³
3. (p⁶)²
4. 3a⁴ × 5a²
5. 18c⁹ ÷ 6c⁵
6. (2b³)⁴

Section 2: Zero and Negative Indices

Simplify the following expressions. Write your answers with positive indices only.

1. 12⁰
2. (4x²y)⁰
3. m^-5
4. 7x^-3
5. (x⁸)(x^-3)
6. (k^-2)^-4
7. 15g³ ÷ 5g^-2

Section 3: Fractional Indices

Evaluate the following numerical expressions.

1. 49^1/2
2. 125^1/3
3. 16^3/4
4. 81^-3/4

Section 4: Putting It All Together

These problems combine multiple rules. Simplify each expression fully, leaving your answers in index form with positive indices.

1. (3x⁴y^-2)³
2. (a^1/2 × a^1/3)
3. (8x⁹)^2/3
4. ¾ (25x⁸ / x^-2)^1/2
5. Express (x² × ³√x) ÷ x^1/6 in the form x^k

Answer Key

Section 1: Basic Simplification

1. x⁵⁺⁴ = x⁹
2. y^10-3 = y⁷
3. p^6×2 = p¹²
4. (3 × 5)a⁴⁺² = 15a⁶
5. (18 ÷ 6)c^9-5 = 3c⁴
6. 2⁴(b³)⁴ = 16b¹²

Section 2: Zero and Negative Indices

1. 1
2. 1 (Anything to the power of 0 is 1)
3. 1/m⁵
4. 7/x³ (The power only applies to the x)
5. x^8-3 = x⁵
6. k^-2×-4 = k⁸
7. (15 ÷ 5)g^{3 - (-2)} = 3g³⁺² = 3g⁵

Section 3: Fractional Indices

1. √49 = 7
2. ³√125 = 5
3. (⁴√16)³ = 2³ = 8
4. 1 / 81^3/4 = 1 / (⁴√81)³ = 1 / 3³ = 1/27

Section 4: Putting It All Together

1. 3³(x⁴)³(y^-2)³ = 27x¹²y^-6 = 27x¹²/y⁶
2. a^{1/2 + 1/3} = a^{3/6 + 2/6} = a^5/6
3. 8^2/3(x⁹)^2/3 = (³√8)²(x^9×2/3) = 2²x⁶ = 4x⁶
4. (25x^{8 - (-2)})^1/2 = (25x¹⁰)^1/2 = 25^1/2(x¹⁰)^1/2 = 5x⁵
5. (x² × x^1/3) ÷ x^1/6 = x^{2 + 1/3} ÷ x^1/6 = x^7/3 ÷ x^1/6 = x^{7/3 - 1/6} = x^{14/6 - 1/6} = x^13/6. So, k = 13/6.