📝 Instructions: Rules for Powers and Roots

In this set of exercises, you will combine all the rules you have learned. Pay close attention to the difference between addition (like terms) and multiplication (adding exponents).

Power rules: Only work if the base number is the same.
Roots: Simplify by looking for perfect squares (4, 9, 16, 25 …).
Check: Use the green expandable fields at the bottom to compare step by step.

Tip: A common mistake is a² + a³ = a⁵. This is not allowed! You can only add powers if the exponents are exactly the same.

💡 View the calculation rules (Cheat sheet)

Powers:

a^p · a^q = a^p+q
a^pa^q = a^p-q
(a^p)^q = a^{p · q}

Roots:

√a · √b = √a · b
√a√b = √ab
3√a + 2√a = 5√a

Block 1: Power Basics

Reduce to a single power:

1) a³ · a⁴

2) x⁷ · x²

3) b⁵ · b · b²

4) p¹⁰p³

5) 2x³ · 4x⁵

Show solutions 1-5

1) a³⁺⁴ = a⁷

2) x⁷⁺² = x⁹

3) b⁵⁺¹⁺² = b⁸

4) p^10-3 = p⁷

5) (2 · 4) · x³⁺⁵ = 8x⁸

Block 2: Parentheses & Powers

Simplify as much as possible:

6) (a³)⁴

7) (2x²)³

8) (b²)⁵b³

9) (ab)⁴ · a²

10) (-3p⁴)²

Show solutions 6-10

6) a^{3 · 4} = a¹²

7) 2³ · (x²)³ = 8x⁶

8) b¹⁰b³ = b⁷

9) a⁴ · b⁴ · a² = a⁶b⁴

10) (-3)² · (p⁴)² = 9p⁸

Block 3: Roots

Simplify or reduce:

11) √3 · √5

12) 2√7 + 5√7

13) √18

14) √50 - √8

15) √2 · √8

Show solutions 11-15

11) √3 · 5 = √15

12) (2+5)√7 = 7√7

13) √9 · 2 = 3√2

14) 5√2 - 2√2 = 3√2

15) √16 = 4