Question

in 8 - 11, apply the properties of operations to multiply.

1. $\frac{1}{2}xcdot\frac{-4}{5}x$
2. $(y - 1.1)(-3y)$
3. $(1.1 - 2.7k)(-3.5k)=(-3.5k)(1.1 - 2.7k)$
4. $\frac{1}{8}w(-1 + w)$

in 12 - 19, use the properties of operations to multiply.

1. $(9.1x)(8x)$
2. $(-\frac{10}{5}f - 1)(-\frac{5}{10}f)$
3. $(1+\frac{3}{4}c)(\frac{4}{3}c)$
4. $(-5.23x)(x + 1.7)$
5. $(9.4a)(-4.9a - 0.5 + 1)$
6. $(13b + 11 - 2b)(-\frac{13}{11}b)$
7. $(-2\frac{1}{3}p)(6-\frac{1}{3}p + 3)$
8. $(m + m)(7.5m)$
9. check for reasonableness paola says that when you apply the distributive property to multiply $(2j + 7)$ and $(-5j)$, the result will have two terms. is she correct? explain.
10. what is the area of this tile? 12.5a in.

Explanation:

Step1: Multiply coefficients and add exponents of like - variables

For (9.1x)(8x), we multiply 9.1 and 8, and since \(x^1\times x^1=x^{1 + 1}=x^2\).
\(9.1\times8x^{1+1}=72.8x^{2}\)

Step2: Apply distributive property for \((1+\frac{3}{4}c)(\frac{4}{3}c)\)

\((1+\frac{3}{4}c)(\frac{4}{3}c)=1\times\frac{4}{3}c+\frac{3}{4}c\times\frac{4}{3}c=\frac{4}{3}c + c^{2}\)

Step3: Apply distributive property for \((- 5.23x)(x + 1.7)\)

\((-5.23x)(x + 1.7)=-5.23x\times x-5.23x\times1.7=-5.23x^{2}-8.891x\)

Step4: Apply distributive property for \((9.4a)(-4.9a - 0.5 + 1)\)

\((9.4a)(-4.9a - 0.5 + 1)=9.4a\times(-4.9a)+9.4a\times(-0.5)+9.4a\times1=-46.06a^{2}-4.7a + 9.4a=-46.06a^{2}+4.7a\)

Step5: Simplify \((13b + 11-2b)(-\frac{13}{11}b)\)

First simplify \(13b + 11-2b = 11b+11\), then \((11b + 11)(-\frac{13}{11}b)=11b\times(-\frac{13}{11}b)+11\times(-\frac{13}{11}b)=-13b^{2}-13b\)

Step6: Apply distributive property for \((-2\frac{1}{3}p)(6-\frac{1}{3}p + 3)\)

\(-2\frac{1}{3}=-\frac{7}{3}\), so \((-\frac{7}{3}p)(6-\frac{1}{3}p + 3)=(-\frac{7}{3}p)\times6-(-\frac{7}{3}p)\times\frac{1}{3}p+(-\frac{7}{3}p)\times3=-14p+\frac{7}{9}p^{2}-7p=\frac{7}{9}p^{2}-21p\)

Step7: Simplify \((m + m)(7.5m)\)

Since \(m + m = 2m\), then \((2m)(7.5m)=2\times7.5m^{1 + 1}=15m^{2}\)

Step8: For 20

Using the distributive property \((2j + 7)(-5j)=2j\times(-5j)+7\times(-5j)=-10j^{2}-35j\), which has two terms. So Paola is correct.

Step9: For 21

Assuming the tile is a square with side length \(s = 12.5a\) inches, the area \(A=s^{2}=(12.5a)^{2}=156.25a^{2}\) square inches

Answer:

1. \(72.8x^{2}\)
2. \(\frac{4}{3}c + c^{2}\)
3. \(-5.23x^{2}-8.891x\)
4. \(-46.06a^{2}+4.7a\)
5. \(-13b^{2}-13b\)
6. \(\frac{7}{9}p^{2}-21p\)
7. \(15m^{2}\)
8. Paola is correct. Using the distributive property \((2j + 7)(-5j)=-10j^{2}-35j\) which has two terms.
9. \(156.25a^{2}\) square inches