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 $x > 0$ 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

Study Guide - Practice Questions