Understand WHY a law works.
This is not a worksheet that skips from the first line to the answer. Each worked problem has a guided animation, a complete static solution, an explicit reason for every transformation, a visual flow diagram and a mathematical power-curve reference.
Stage 1 · Recognize
Find the base and exponent. Decide whether you are multiplying, dividing, raising a power, or taking a root.
Stage 2 · Check the domain
Zero denominators and even roots of negative numbers are not allowed in ordinary real-number calculations.
Stage 3 · Transform
Use one justified law at a time. Read each highlighted explanation before advancing.
Stage 4 · Verify
Substitute numbers or reverse the operation. Compare the original expression with the final answer.
Choose a starting lesson
Five reading habits that prevent mistakes
Base
In 3⁴, the base is 3. In (2x)⁴, the whole product 2x is the base.
Exponent
In 3⁴, the exponent is 4: four factors of 3.
Negative exponent
In 2⁻³, the minus sign instructs a reciprocal: 1/8, not −8.
Fractional exponent
In 8²⁄³, the top 2 is the POWER and bottom 3 is the ROOT.
Principal root
√16 = 4, not ±4. When solving x²=16, both x=4 and x=−4 may occur.
In the lessons, “real-number domain” means we explain exactly when each fractional power or reciprocal is allowed.
Combine laws without losing the logic.
Use the L2 → L5 progression. Every problem is solved as a sequence of justified transformations; difficult examples include domain restrictions and special root/absolute-value traps.
Common errors worth teaching explicitly
English pronunciation guide
√a
“The square root of a.”
∛a
“The cube root of a.”
ⁿ√a
“The nth root of a.” (en-th)
a¹⁄ⁿ
“a raised to the power of one over n.”
aᵐ⁄ⁿ
“a to the power of m over n.”
a⁻³
“a to the negative third power.”
All 50 problem solutions and every visualization are included in this single HTML file. No external libraries or internet connection are needed. The reference function graph illustrates a representative power and does not replace the algebraic proof.