Laws of exponents

In exponent notation $a^n$, the base is the number being multiplied repeatedly, and the exponent is the small raised number that tells you how many times.

In $7^4$, the large number 7 is being multiplied by itself, so 7 is the base. The small raised number 4 tells you how many times 7 is multiplied, so 4 is the exponent.

The correct identification is Base: 7, Exponent: 4. This means $7^4=7\times 7\times 7\times 7$.

We need to correctly label the base and exponent in $7^4$. The base is always the larger number written normally, and the exponent is the small raised number.

This option swaps the base and exponent. The 4 is the small raised number, so it is the exponent — not the base. Eliminate this option.

Neither 'Base: 28, Exponent: 4' nor 'Base: 7, Exponent: 28' is correct. The value 28 comes from $7\times 4=28$, which is a common mistake of multiplying instead of using the exponent notation. Neither 28 appears as the base or exponent in $7^4$.

The only remaining option is Base: 7, Exponent: 4, which correctly identifies 7 as the repeated number and 4 as the count of repetitions.

When multiplying algebraic terms, multiply the numbers (coefficients) together and apply the Product Rule to the variables with the same base.

$3\times 5=15$

Both terms have base $a$. Using the Product Rule: $a^2\times a^3=a^{2+3}=a^5$.

$3a^2\times 5a^3=15a^5$

Multiply the numbers: $3\times 5=15$. Add the exponents of $a$: $2+3=5$. So the answer should be $15a^5$.

This option multiplies the exponents ($2\times 3=6$) instead of adding them. The Product Rule says to add exponents when multiplying same bases: $a^2\times a^3=a^{2+3}=a^5$, not $a^6$.

This option adds the coefficients ($3+5=8$) instead of multiplying them. When multiplying terms, the coefficients must be multiplied: $3\times 5=15$, not 8.

This is not a valid simplification at all. Multiplication of two terms gives a single term, not a sum. This option makes no algebraic sense here.

$15a^5$ is correct: coefficient $3\times 5=15$, and exponent $2+3=5$.