Question

1. (1.73) multiply or divide (as indicated) the fractions and select the simplified answer: $\frac{2y^{4}+6y^{2}}{15y}cdot\frac{8y}{4y^{3}+12y}$

a) $\frac{16y^{7}}{31}$
b) $\frac{2y^{4}+6y^{2}+8y}{4y^{3}+27y}$
c) $\frac{8y}{15}$
d) $\frac{2y^{2}}{3}$
e) $\frac{4y}{15}$

1. (1.79) select the fully simplified expression for the area of the rectangle.

a) $x^{2}-2x - 1$
b) $2x$
c) $x^{2}-1$
d) $x^{2}+2x + 1$
e) $x^{2}+1$

1. (1.79) select the fully simplified expression for the area of the rectangle.

a) $3x^{2}$
b) $3.5x$
c) $7x$
d) $3.5x^{2}$
e) $7x^{2}$

Explanation:

Step1: Simplify the first - fraction in problem 14

Factor the numerator of $\frac{2y^{4}+6y^{2}}{15y}$: $2y^{4}+6y^{2}=2y^{2}(y^{2} + 3)$. So, $\frac{2y^{4}+6y^{2}}{15y}=\frac{2y^{2}(y^{2}+3)}{15y}=\frac{2y(y^{2}+3)}{15}$.

Step2: Simplify the second - fraction in problem 14

Factor the denominator of $\frac{8y}{4y^{3}+12y}$: $4y^{3}+12y = 4y(y^{2}+3)$. So, $\frac{8y}{4y^{3}+12y}=\frac{8y}{4y(y^{2}+3)}=\frac{2}{y^{2}+3}$.

Step3: Multiply the two simplified fractions

$\frac{2y(y^{2}+3)}{15}\times\frac{2}{y^{2}+3}=\frac{4y}{15}$.

Step4: Solve problem 15

The area of a rectangle is given by $A = l\times w$. Here, $l=x + 1$ and $w=x - 1$. Using the difference - of - squares formula $(a + b)(a - b)=a^{2}-b^{2}$, we have $(x + 1)(x - 1)=x^{2}-1$.

Step5: Solve problem 16

The area of a rectangle with length $1.5x$ and width $2x$ is $A=1.5x\times2x=(1.5\times2)x^{1 + 1}=3x^{2}$.

Answer:

1. e) $\frac{4y}{15}$
2. c) $x^{2}-1$
3. a) $3x^{2}$