Simplify expressions with the laws of integral indices

Follow this one with me, a step at a time. Try to guess each step first.

Worked example 1

Simplify (x²)⁴ × x³ ÷ x⁴ and express the answer with positive indices.

Remember: Same base: multiply the indices for a power of a power, add them for a product, subtract them for a quotient.

A power of a power multiplies the indices: (x²)⁴ = x⁸.
Add the index of the product and subtract the index of the quotient: 8 + 3 − 4, which is 7.
The answer is x⁷.

Common mistakes

Added the indices in a division. Dividing powers of the same base means subtracting them.
Multiplied the indices when multiplying powers of the same base. They should be added.
Added the indices in a power of a power. There the indices are multiplied.