Laws of exponents — all laws

We are multiplying two powers with the same base (5). This calls for the Product Rule: $a^m\times a^n=a^{m+n}$.

Keep the base as 5 and add the exponents: $5^3\times 5^6=5^{3+6}=5^9$

The simplified expression is $5^9$. The base stays as 5 — it does not get multiplied.

We need to simplify $5^3\times 5^6$ using exponent laws. The key question is: what happens to the base and exponents when multiplying same-base powers?

The option $5^{18}$ would result from multiplying the exponents ($3\times 6=18$). That rule applies to a power of a power like $(5^3{)}^6$, not a product. So this is wrong.

The option $25^9$ multiplies both the base ($5\times 5=25$) and adds the exponents. This is a double error — the base must stay as 5, not change to 25.

The option $5^{15}$ does not match any standard rule here. $3+6=9$, not 15. This value has no logical basis.

The correct answer is $5^9$, found by applying the product rule: $5^3\times 5^6=5^{3+6}=5^9$.

The expression $\frac{7^{10}}{7^4}$ is a division (a fraction) of two powers with the same base. This matches the Quotient Rule.

The Quotient Rule states $\frac{a^m}{a^n}=a^{m-n}$. So: $\frac{7^{10}}{7^4}=7^{10-4}=7^6$

The student subtracted the exponents ($10-4=6$), which is exactly the Quotient Rule.

We need to identify which law produces the result $7^6$ from $\frac{7^{10}}{7^4}$.

The Product Rule applies to multiplication, giving $a^{m+n}$. Here we have division, not multiplication, so this law does not apply.

The Power of a Power Rule applies to expressions like $(7^4{)}^{10}$, where a power is raised to another power. This is a division expression, so that rule is irrelevant.

The Zero Exponent Rule tells us that $a^0=1$. Since the result is $7^6$, not 1, this rule is not what was used.

The correct law is the Quotient Rule: $\frac{a^m}{a^n}=a^{m-n}$, giving $7^{10-4}=7^6$.