Question

1. essential question how can you use the properties of operations to multiply linear expressions?
2. check for reasonableness jayden multiplies -4y·1/5y, and gets an answer of -4/5y. how can you tell without multiplying that his answer is incorrect?
3. choose efficient methods explain the method you would use to multiply the expression below. 11/17×(1/5x + 3/5+8/10x + 12/5)

in 4 - 6, use the properties of operations to multiply.

1. (2/5·b)·b
2. (1.2 + p)(-9.8p)
3. 3.6(z - 2.3)
4. a 3.5 inch - by - 5 inch image is being enlarged by a scale factor of x. express the area of the enlarged image in terms of x.

Explanation:

Step1: Simplify \((\frac{2}{5}\cdot b)\cdot b\)

Use the associative property of multiplication \((a\cdot b)\cdot c=a\cdot(b\cdot c)\) and the rule of exponents \(a^m\cdot a^n=a^{m + n}\). Here \(a = b\), \(m = 1\), \(n=1\), so \((\frac{2}{5}\cdot b)\cdot b=\frac{2}{5}\cdot(b\cdot b)=\frac{2}{5}b^{2}\).

Step2: Simplify \((1.2 + p)(-9.8p)\)

Use the distributive property \(a(b + c)=ab+ac\), where \(a=-9.8p\), \(b = 1.2\), \(c = p\). Then \((1.2 + p)(-9.8p)=1.2\times(-9.8p)+p\times(-9.8p)=-11.76p-9.8p^{2}\).

Step3: Simplify \(3.6(z - 2.3)\)

Use the distributive property \(a(b - c)=ab - ac\), where \(a = 3.6\), \(b=z\), \(c = 2.3\). So \(3.6(z - 2.3)=3.6z-3.6\times2.3=3.6z - 8.28\).

Step4: Analyze Jayden's error

When multiplying \(-4y\cdot\frac{1}{5}y\), we should use the rule \(a^m\cdot a^n=a^{m + n}\) and \(a\times b\) for coefficients. The correct result is \((-4\times\frac{1}{5})y^{1 + 1}=-\frac{4}{5}y^{2}\), not \(-\frac{4}{5}y\). Jayden didn't apply the exponent - rule for multiplying variables correctly.

Step5: Simplify \(\frac{11}{17}\times(\frac{1}{5}x+\frac{3}{5}+\frac{8}{10}x+\frac{12}{5})\)

First, combine like - terms inside the parentheses: \(\frac{1}{5}x+\frac{8}{10}x=\frac{1}{5}x+\frac{4}{5}x=\frac{1 + 4}{5}x=x\), and \(\frac{3}{5}+\frac{12}{5}=\frac{3 + 12}{5}=3\). So the expression becomes \(\frac{11}{17}(x + 3)\). Then use the distributive property: \(\frac{11}{17}(x + 3)=\frac{11}{17}x+\frac{11\times3}{17}=\frac{11}{17}x+\frac{33}{17}\).

Step6: Find the area of the enlarged image

The original area of the image is \(A_0=3.5\times5 = 17.5\) square inches. When enlarged by a scale factor of \(x\), the new dimensions are \(3.5x\) inches and \(5x\) inches. The area of the enlarged image \(A=(3.5x)\times(5x)=17.5x^{2}\) square inches.

Answer:

1. \(\frac{2}{5}b^{2}\)
2. \(-11.76p-9.8p^{2}\)
3. \(3.6z - 8.28\)
4. Jayden didn't apply the exponent - rule for multiplying variables correctly. The correct result should be \(-\frac{4}{5}y^{2}\), not \(-\frac{4}{5}y\).
5. First combine like - terms inside the parentheses, then use the distributive property. The result is \(\frac{11}{17}x+\frac{33}{17}\).
6. \(17.5x^{2}\) square inches.