Question

24 simplify. your answer should contain only positive exponents.\\(\frac{x^{15}y}{x^{10}}\\)\
25 simplify. your answer should contain only positive exponents.\\(\left(\frac{x^2y^4}{x^3y^4}\
ight)^0\\)\
26 four tile patterns (a, b, c, and d) are represented by the lines graphed below\
which pattern has the fastest growth rate?\
graph for question 26\
27 four tile patterns (a, b, c and d) are represented by the lines graphed below.\
which pattern has the slowest growth rate?\
graph for question 27\
28 calculate the slope of the line segment below given two points \\((-4, 4)\\) and \\((2, -5)\\).\
hint: \\(m = \frac{\delta y}{\delta x}\\)\
29 find the equation of the line segment below\
in \\(y = mx + b\\) form given two points\
\\((-4, 4)\\) and \\((2, -5)\\).\
hint: \\(m = \frac{\delta y}{\delta x}\\)

Explanation:

Problem 24

Step1: Apply exponent rule for division

When dividing terms with the same base, subtract the exponents: $\frac{x^{a}}{x^{b}} = x^{a - b}$. Here, for the $x$ terms: $\frac{x^{15}}{x^{10}} = x^{15 - 10}$.

Step2: Simplify the exponents

$15 - 10 = 5$, so $\frac{x^{15}y}{x^{10}} = x^{5}y$.

Step1: Apply exponent rule for division inside the parentheses

First, simplify $\frac{x^{2}y^{4}}{x^{3}y^{4}}$. For $x$ terms: $\frac{x^{2}}{x^{3}} = x^{2 - 3}=x^{-1}$, for $y$ terms: $\frac{y^{4}}{y^{4}} = y^{4 - 4}=y^{0}=1$ (since any non - zero number to the power of 0 is 1). So $\frac{x^{2}y^{4}}{x^{3}y^{4}}=\frac{x^{-1}\times1}{1}=x^{-1}$.

Step2: Apply the zero - exponent rule

Now we have $(x^{-1})^{0}$. According to the rule $(a^{m})^{n}=a^{mn}$, so $(x^{-1})^{0}=x^{-1\times0}=x^{0}=1$ (since any non - zero number to the power of 0 is 1).

Step1: Recall the relationship between slope and growth rate

The growth rate of a linear pattern (represented by a line) is related to the slope of the line. The steeper the line (the larger the slope), the faster the growth rate.

Step2: Analyze the slopes of the lines

Looking at the graph for problem 26, line D is the steepest among lines A, B, C, and D. A steeper line has a larger slope, which means a faster growth rate.

Answer:

$x^{5}y$

Problem 25