All real $$x$$ and $$y$$ such that $$x^{3y}$$ \neq 2$$

Question 1

Simplify the following expression: $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$. Which of the following is the correct simplified form?

Accepted Answer

$$\frac{x$$^{5y}$$ + 30}{x$$^{15y}$$ - 10}$$

Question 2

If $$x = 2$$ and $$y = 3$$, what is the value of $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

$$\left( \frac{8 + 6}{512 - 2} \$$right)^5$$ = \left( \frac{14}{510} \$$right)^5$$

Question 3

Which property of exponents is used to simplify $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

Power of a quotient: $$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$

Question 4

Given the expression $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$, which of the following is a valid step in its simplification?

Accepted Answer

Raise both numerator and denominator to the 5th power: $$\frac{(x^y + 6)^5$}{(x$$^{3y}$$ - 2)^5$}$$

Question 5

If $$x = 1$$ and $$y = 2$$, what is the value of $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

$$\left( \frac{1^2$ + 6}{1$$^{6}$$ - 2} \$$right)^5$$ = \left( \frac{7}{-1} \$$right)^5$$ = -16807$$

Question 6

Which of the following is equivalent to $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$ for x$$ \u003e 0$$ and y$ an integer?

Accepted Answer

$$\frac{(x^y + 6)^5$}{(x$$^{3y}$$ - 2)^5$}$$

Question 7

If $$x^y + 6 = 0$$, what is the value of $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

$$0$$

Question 8

Which of the following best describes the domain of $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

All real $$x$$ and $$y$$ such that $$x^{3y}$$ 
eq 2$$

Question 9

If $$x = -2$$ and $$y = 1$$, what is the value of $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

$$\left( \frac{-2 + 6}{-8 - 2} \$$right)^5$$ = \left( \frac{4}{-10} \$$right)^5$$ = \left( -0.4 \$$right)^5$$ = -0.01024$$

Question 10

Which of the following is NOT a valid simplification of $$\left( \frac{x^y + 6}{x$$^{3y}$$ - 2} \$$right)^5$$?

Accepted Answer

$$\frac{x^y + 6}{x$$^{3y}$$ - 2}$$

Properties of Exponents and Simplifying Algebraic Expressions

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