Q: Which of the following steps is NOT correct when simplifying $$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$?

Multiplying exponents for $$y$$: $$4 \times (-6) = -24$$

Question 1

Simplify the following exponential expression:

$$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$

Which of the following is the correct simplified form?

Accepted Answer

$$\displaystyle \frac{10y$$^{10}$$}{9x^2$}$$

Question 2

Which of the following steps is NOT correct when simplifying $$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$?

Accepted Answer

Multiplying exponents for $$y$$: $$4 	imes (-6) = -24$$

Question 3

If $$x$$ and $$y$$ are both positive real numbers, what is the value of $$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$ when $$x=3$$ and $$y=2$$?

Accepted Answer

$$\displaystyle \frac{10240}{81}$$

Question 4

Which property of exponents is used to simplify $$y^4$$ \div $$y^{-6}$$?

Accepted Answer

$$a^m \div a^n = a$$^{m-n}$$, and since -(-6) = +6$, 4-(-6)=10$

Question 5

Which of the following is equivalent to $$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$ with only positive exponents?

Accepted Answer

$$\displaystyle \frac{10y$$^{10}$$}{9x^2$}$$

Question 6

A biologist is modeling the growth of a bacteria population using the formula $$P = \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$, where $$x$$ and $$y$$ are environmental factors. If $$x$$ doubles and $$y$$ triples, by what factor does $$P$$ change?

Accepted Answer

$$\displaystyle \frac{10 \cdot (2)^2$ \cdot (3$$)^{10}$$}{9 \cdot (2)^2$}$$, so $$P$$ increases by $$3^{10}$$ times

Question 7

Which of the following is the correct step to simplify $$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$}$$?

Accepted Answer

Rewrite as $$\displaystyle \frac{30}{27} \cdot x$$^{2-4}$$ \cdot y$$^{4-(-6)}$$

Question 8

If $$\displaystyle \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$} = kx^a y^b$$, what are the values of $$k$$, $$a$$, and $$b$$?

Accepted Answer

$$k=\frac{10}{9},\ a=-2,\ b=10$$

Question 9

Which of the following is the correct way to express $$x^{-2}$$ using only positive exponents?

Accepted Answer

$$\displaystyle \frac{1}{x^2$}$$

Question 10

A chemist uses the formula $$C = \frac{30x^2$y^4$$}{$$27x^4y$$^{-6}$$} to $$calculate the concentration of a solution. If $$x=1$$ and $$y=10$$, what is the value of $$C$$?

Accepted Answer

$$\displaystyle \frac{10 \cdot 10$$^{10}$$}{9}$$

Simplifying Exponential Expressions

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