Simplify the following expressions: (a) x * 9x (b) 9bz * b (e) d^4 / (cd^2) (f) k^2*p * 8p^2*k^3 (a) c * 9x * 2xc (c) (3a^3b)^2 (e) -3/8 * (2a^2 / ab) (g) 7wq * (11q^2 + 15hw)

Steps to Solve

7a. Simplify $x \cdot 9x$ Multiply the coefficients and combine the $x$ terms:

$x \cdot 9x = 9 \cdot x \cdot x = 9x^2$

7b. Simplify $9bz \cdot b$ Multiply the $b$ terms:

$9bz \cdot b = 9 \cdot b \cdot b \cdot z = 9b^2z$

7e. Simplify $\frac{d^4}{cd^2}$ Divide the $d$ terms:

$\frac{d^4}{cd^2} = \frac{d^4}{d^2} \cdot \frac{1}{c} = d^{4-2} \cdot \frac{1}{c} = \frac{d^2}{c}$

7f. Simplify $k^2p \cdot 8p^2k^3$ Multiply the coefficients and combine like terms:

$k^2p \cdot 8p^2k^3 = 8 \cdot k^2 \cdot k^3 \cdot p \cdot p^2 = 8k^{2+3}p^{1+2} = 8k^5p^3$

8a. Simplify $c \cdot 9x \cdot 2xc$ Multiply the coefficients and combine like terms:

$c \cdot 9x \cdot 2xc = 9 \cdot 2 \cdot c \cdot c \cdot x \cdot x = 18c^2x^2$

8c. Simplify $(3a^3b)^2$ Apply the power to each term inside the parentheses:

$(3a^3b)^2 = 3^2 \cdot (a^3)^2 \cdot b^2 = 9 \cdot a^{3 \cdot 2} \cdot b^2 = 9a^6b^2$

8e. Simplify $-\frac{3}{8}x \cdot (2a^2 \div ab)$ Simplify inside the parentheses first. Note that $2a^2 \div ab = \frac{2a^2}{ab} = \frac{2a}{b}$. Then, multiply by $-\frac{3}{8}x$:

$-\frac{3}{8}x \cdot \frac{2a}{b} = -\frac{3 \cdot 2}{8} \cdot \frac{ax}{b} = -\frac{6}{8} \cdot \frac{ax}{b} = -\frac{3}{4} \cdot \frac{ax}{b} = -\frac{3ax}{4b}$

8g. Simplify $7wq \cdot (11q^2 + 15hw)$ Distribute the $7wq$ term:

$7wq \cdot (11q^2 + 15hw) = 7wq \cdot 11q^2 + 7wq \cdot 15hw = 7 \cdot 11 \cdot w \cdot q \cdot q^2 + 7 \cdot 15 \cdot w \cdot q \cdot h \cdot w = 77wq^3 + 105hw^2q$