📝 Instructions: Solving Power Functions

In this assignment, you will practice with power functions of the form y = a · xⁿ. The most important thing is to recognize whether the exponent is even or odd.

Even exponent (x², x⁴, …): Pay attention! There are often two solutions (positive and negative).
Odd exponent (x³, x⁵, …): There is always exactly one solution.
Standard form: Make sure the power (xⁿ) stands alone before you take the root.

Tip: An even power can never have a negative result. Does it say x⁴ = -16? Then there are no solutions.

💡 Calculation Rule: Number of solutions

When solving xⁿ = c:

n is EVEN:
- c > 0 → 2 solutions: $x = \sqrt[n]{c}$ and $x = - \sqrt[n]{c}$
- c < 0 → no solutions
n is ODD:
- Always 1 solution: $x = \sqrt[n]{c}$

Block 1: Basic Solving

1) x³ = 27

2) x⁴ = 16

3) x⁵ = -32

4) x⁶ = 1

5) x⁴ = -81

Show solutions

1) x =

\sqrt[3]{27}

= 3

2) x = 2 or x = -2 (even power!)

3) x =

\sqrt[5]{- 32}

= -2

4) x = 1 or x = -1
5) No solution (even power cannot be negative)

Block 2: With factors

6) 2x³ = 54

7) 3x⁴ = 243

8) 0.5x³ = 500

9) 5x⁴ + 20 = 100

10) -2x⁵ = 64

Show solutions

6) x³ = 27 → x = 3
7) x⁴ = 81 → x = 3, x = -3
8) x³ = 1000 → x = 10
9) 5x⁴ = 80
x⁴ = 16 → x=2, x=-2
10) x⁵ = -32 → x = -2

Block 3: Critical Thinking

11) x³ = -2

12) x² + 5 = 9

13) x² = x

14) 2x³ + 6 = 0

15) x⁴ + 8 = 5

Show solutions

11) No solution (root cannot be negative)
12) x² + 5 = 9 → x² = 4 → x=2, x=-2
13) x = x² → x² - x = 0 → x=0, x=1
14) 2x+6 = (x+3)² → x²+4x+3=0 → x=-1, x=-3
15) x⁴ = -3 → No solution