Two core rules, both relying on the base being identical:

Multiplying adds the indices, dividing subtracts them. You can see why by expanding: $a^3 \times a^4 = (a\times a\times a)\times(a\times a\times a\times a) = a^7$. So $k^5 \times k^7 = k^{12}$, and $p^3 \div p^{10} = p^{-7}$.

Here you multiply the indices rather than add them — a completely different operation to the multiplication rule above, which is exactly why the two get confused. $(x^2)^3 = x^2 \times x^2 \times x^2 = x^{2+2+2} = x^6$, confirming $2 \times 3 = 6$.

Numbers and letters are processed completely separately. For $4a^3 \times 7a^2$: multiply the coefficients ($4\times7=28$), then apply the index law to the letters ($a^{3+2}=a^5$), giving $28a^5$. A very common slip is to multiply the coefficients correctly but then also multiply the indices (getting $28a^6$) — remember, multiplying terms together always means adding their indices.

When a power sits outside a bracket, it applies to everything inside, including the coefficient: $(2x^3)^3 = 2^3 \times (x^3)^3 = 8x^9$, not $2x^9$.

The zero rule follows directly from the division law: $a^m \div a^m = a^{m-m} = a^0$, but anything divided by itself is $1$. So $5 \times 3^0 = 5 \times 1 = 5$ — the power applies only to the base right next to it, not the whole expression.

The negative sign is a position instruction (take the reciprocal), never a sign on the value itself: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$, which is positive. A negative base raised to a negative power never becomes a negative number just because of the index.

Spot a shared prime base hiding inside different numbers. To simplify $4^3 \times 8^2$ as a single power of $2$: rewrite $4 = 2^2$ and $8=2^3$, apply the power-of-a-power rule to get $2^6 \times 2^6$, then add the indices: $2^{12}$.

Simplify $(3y^2)^2 \times 2y^5$, and simplify $5p^7 \div 100p^3$ fully.

Part 1:

1. Resolve the bracket first: $(3y^2)^2 = 3^2 \times y^{2\times2} = 9y^4$.
2. Multiply by the remaining term: $9y^4 \times 2y^5$.
3. Coefficients: $9\times2=18$. Indices: $4+5=9$.
4. Result: $18y^9$.

Part 2:

1. Coefficients: $5 \div 100 = \frac{1}{20}$.
2. Indices: $7-3=4$, giving $p^4$ on top — but wait, check direction: it's $5p^7 \div 100p^3 = \frac{5}{100}p^{7-3}=\frac{1}{20}p^4=\frac{p^4}{20}$.

Always double-check which number is divided by which before committing to the sign of the resulting index.