Question

simplify:

1. (a) \\(\frac{x^3 + 8x^2 + 15x}{x^2 - 4x + 4} \cdot \frac{x^2 + 2x - 8}{12x + x^3 + 7x^2}\\)
2. (b) \\(32^{-\frac{2}{5}}\\)
3. (c) \\(\frac{a^2x - \frac{a}{x}}{ax - \frac{4}{x}}\\)
4. (d) \\(\frac{(21,000 \times 10^{-42})(500,000)}{0.00015 \times 10^{-7}}\\)

Explanation:

Step1: Factor numerator of first fraction

$x^3 + 8x^2 + 15x = x(x^2 + 8x + 15) = x(x+3)(x+5)$

Step2: Factor denominator of first fraction

$x^2 - 4x + 4 = (x-2)^2$

Step3: Factor numerator of second fraction

$x^2 + 2x - 8 = (x+4)(x-2)$

Step4: Factor denominator of second fraction

$x^3 + 7x^2 + 12x = x(x^2 + 7x + 12) = x(x+3)(x+4)$

Step5: Rewrite product and cancel terms

$\frac{x(x+3)(x+5)}{(x-2)^2} \cdot \frac{(x+4)(x-2)}{x(x+3)(x+4)} = \frac{x+5}{x-2}$

Step1: Rewrite exponent as root

$32^{-2/5} = \frac{1}{32^{2/5}} = \frac{1}{(32^{1/5})^2}$

Step2: Calculate 5th root of 32

$32^{1/5} = 2$

Step3: Square the result and take reciprocal

$\frac{1}{2^2} = \frac{1}{4}$

Step1: Factor numerator

$a^2x - a = a(ax - 1)$

Step2: Simplify denominator fraction

$ax - \frac{4}{x} = \frac{ax^2 - 4}{x}$

Step3: Rewrite division as multiplication

$\frac{a(ax - 1)}{\frac{ax^2 - 4}{x}} = a(ax - 1) \cdot \frac{x}{ax^2 - 4}$

Step4: Factor denominator quadratic

$ax^2 - 4 = (\sqrt{a}x - 2)(\sqrt{a}x + 2)$ (Note: If $a$ is a perfect square, $ax^2-4=(mx-2)(mx+2)$ where $m^2=a$; in general form, the simplified expression is $\frac{ax(ax - 1)}{ax^2 - 4}$)

Step1: Rewrite numbers in scientific notation

$21,000 = 2.1 \times 10^4$; $500,000 = 5 \times 10^5$; $0.00015 = 1.5 \times 10^{-4}$

Step2: Multiply numerator terms

$(2.1 \times 10^4 \times 10^{-42})(5 \times 10^5) = (2.1 \times 5) \times 10^{4-42+5} = 10.5 \times 10^{-33} = 1.05 \times 10^{-32}$

Step3: Divide by denominator

$\frac{1.05 \times 10^{-32}}{1.5 \times 10^{-4} \times 10^{-7}} = \frac{1.05}{1.5} \times 10^{-32+4+7} = 0.7 \times 10^{-21} = 7 \times 10^{-22}$

Answer:

1. $\boldsymbol{\frac{x+5}{x-2}}$
2. $\boldsymbol{\frac{1}{4}}$
3. $\boldsymbol{\frac{ax(ax - 1)}{ax^2 - 4}}$
4. $\boldsymbol{7 \times 10^{-22}}$