Here are the step-by-step solutions for each problem on the worksheet. The main rule used here is the
Product Rule for Exponents: when multiplying terms with the same base, you add their exponents ($a^m \cdot a^n = a^{m+n}$). You also multiply the coefficients (numbers in front) normally.
1) $9^4 \cdot 9^3$
* Same base (9), so add exponents: $4 + 3 = 7$.
* Answer: $9^7$
2) $(\frac{1}{w})^6 \cdot (\frac{1}{w})^2$
* Same base ($\frac{1}{w}$), so add exponents: $6 + 2 = 8$.
* Answer: $(\frac{1}{w})^8$ or $\frac{1}{w^8}$
3) $r^3s^2 \cdot 8r^3s^4 \cdot 4rs^5$
* Multiply coefficients: $1 \cdot 8 \cdot 4 = 32$.
* Add $r$ exponents: $3 + 3 + 1 = 7$.
* Add $s$ exponents: $2 + 4 + 5 = 11$.
* Answer: $32r^7s^{11}$
4) $7y \cdot 2y^2$
* Multiply coefficients: $7 \cdot 2 = 14$.
* Add $y$ exponents: $1 + 2 = 3$ (remember $y$ is $y^1$).
* Answer: $14y^3$
5) $(\frac{1}{5})^4 \cdot (\frac{1}{5})^3 \cdot (\frac{1}{5})^2$
* Same base ($\frac{1}{5}$), add exponents: $4 + 3 + 2 = 9$.
* Answer: $(\frac{1}{5})^9$ or $\frac{1}{5^9}$
6) $5d^3n^6 \cdot 4dn^4$
* Multiply coefficients: $5 \cdot 4 = 20$.
* Add $d$ exponents: $3 + 1 = 4$.
* Add $n$ exponents: $6 + 4 = 10$.
* Answer: $20d^4n^{10}$
7) $8nr^4 \cdot 9n^6r^3$
* Multiply coefficients: $8 \cdot 9 = 72$.
* Add $n$ exponents: $1 + 6 = 7$.
* Add $r$ exponents: $4 + 3 = 7$.
* Answer: $72n^7r^7$
8) $(\frac{1}{k})^3 \cdot (\frac{1}{k})^4 \cdot (\frac{1}{k})^5$
* Same base ($\frac{1}{k}$), add exponents: $3 + 4 + 5 = 12$.
* Answer: $(\frac{1}{k})^{12}$ or $\frac{1}{k^{12}}$
9) $5z^5 \cdot 6z^{-3} \cdot 3z^6$
* Multiply coefficients: $5 \cdot 6 \cdot 3 = 90$.
* Add $z$ exponents: $5 + (-3) + 6 = 8$.
* Answer: $90z^8$
10) $9 \cdot 9^{-5}$
* Same base (9), add exponents: $1 + (-5) = -4$.
* Result: $9^{-4}$. To make the exponent positive, move it to the denominator.
* Answer: $\frac{1}{9^4}$
11) $9b \cdot 6b^3$
* Multiply coefficients: $9 \cdot 6 = 54$.
* Add $b$ exponents: $1 + 3 = 4$.
* Answer: $54b^4$
12) $9s^4 \cdot 7s^{-2}$
* Multiply coefficients: $9 \cdot 7 = 63$.
* Add $s$ exponents: $4 + (-2) = 2$.
* Answer: $63s^2$
13) $2d^3c^6 \cdot 8d^6c^2$
* Multiply coefficients: $2 \cdot 8 = 16$.
* Add $d$ exponents: $3 + 6 = 9$.
* Add $c$ exponents: $6 + 2 = 8$.
* Answer: $16d^9c^8$
14) $s^2 \cdot s^6$
* Same base ($s$), add exponents: $2 + 6 = 8$.
* Answer: $s^8$
Final Answer:
1) $9^7$
2) $\frac{1}{w^8}$
3) $32r^7s^{11}$
4) $14y^3$
5) $\frac{1}{5^9}$
6) $20d^4n^{10}$
7) $72n^7r^7$
8) $\frac{1}{k^{12}}$
9) $90z^8$
10) $\frac{1}{9^4}$
11) $54b^4$
12) $63s^2$
13) $16d^9c^8$
14) $s^8$
Parent Tip: Review the logic above to help your child master the concept of exponents worksheets grade 8.
