Multiple Choice

1. Simplify the expression: $(\frac{12x^5 y^{-3}}{3x^{-1} y^2})^{-2}$

   $\frac{y^{10}}{16x^{12}}$

   $\frac{y^2}{16x^8}$

   $16x^{12}y^{10}$

   $\frac{16x^{12}}{y^{10}}$

2. Simplify the expression: $(27a^9 b^{-3})^{1/3} \cdot (4a^{-2} b^4)^{1/2}$

   $5a^2 b$

   $6a^4 b^3$

   $54a^{11/3} b^{1/3}$

   $6a^2 b$

3. Simplify the expression, assuming $p \neq 0$ and $q \neq 0$: $\frac{[(p^3 q^{-2})^{-1} \cdot p^0]^3}{(p^{-2} q^4)^2}$

   $\frac{1}{p^5 q^2}$

   $\frac{p^5}{q^2}$

   $p^{13} q^{10}$

   $\frac{q^2}{p^5}$

4. Simplify the expression: $81^{-1/4} \cdot \sqrt[3]{(x^6 y^3)^{-2}}$

   $\frac{1}{3x^4 y^2}$

   $3x^4 y^2$

   $\frac{x^4 y^2}{3}$

   $\frac{1}{27x^4 y^2}$

5. Simplify the expression: $(\frac{a^{-2} b^3 c^{-1}}{a^4 b^{-1} c^2})^3 \cdot (\frac{a^0 b^2}{c^{-3}})^{-1}$

   $\frac{b^{14}}{a^{18} c^{6}}$

   $\frac{a^{24}}{b^{14} c^{15}}$

   $a^{18} b^{10} c^{12}$

   $\frac{b^{10}}{a^{18} c^{12}}$

6. Simplify the expression: $\frac{(2x^2 y^{-3})^{-3}}{(4x^{-1} y^2)^2}$

   $\frac{y^5}{128x^4}$

   $\frac{y^{13}}{32x^8}$

   $\frac{128x^4}{y^5}$

   $\frac{y^5}{2x^4}$

7. If $9^{3x+1} = (27)^{x-2}$, what is the value of $x$?

   $x = 4$

   $x = -8/3$

   $x = -4/3$

   $x = 8/3$

8. Find the value of $\left[ \left( \frac{1}{64} \right)^{-2/3} \right]^{1/2}$

   $1/4$

   $64$

   $16$

   $4$

9. If $A = 3x^{-1}$ and $B = \frac{9x^2}{y}$, simplify the expression $(AB^2)^0 \cdot A^2$.

   $\frac{9}{x^2}$

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

   $9x^4$

   $\frac{81x^2}{y^2}$

10. Simplify the expression $\frac{x^n y^{2n+1}}{(x^{1-n} y^n)^{-2}}$

    $\frac{x^n y^{4n+1}}{x^2}$

    $x^{3n-2} y$

    $x^{2-n} y$

    $x^{2-n} y^{4n+1}$

11. Simplify the expression: $27^{-2/3} \cdot \frac{1}{\sqrt{9x^4}}$.

    $1/(243x^2)$

    $3/x^2$

    $1/(27x^2)$

    $1/(27x^4)$

12. Simplify the expression: $(\frac{16a^{-8}}{b^4})^{3/4}$.

    $8/(a^6 b^3)$

    $12/(a^6 b^3)$

    $1/(4a^6 b^3)$

    $8b^3/a^6$

13. Simplify the expression: $\sqrt[3]{\frac{64x^{12}y^{-6}}{(2x^{-1}y)^3}}$.

    $8x^{15}/y^9$

    $2x^5/y^3$

    $2x^5y^3$

    $2x^3/y^3$

14. Simplify the expression: $(32)^{2/5} \cdot \frac{16^3 \cdot 2^{-4}}{4^{-1}}$.

    $1$

    $256$

    $8192$

    $4096$

15. Simplify the expression: $\frac{(x^{-1}y^4)^2}{\sqrt{16x^8y^{-2}}}$.

    $y^9/(4x^2)$

    $y^9/(4x^6)$

    $y^9/(8x^6)$

    $y^7/(4x^6)$

16. Consider the formal exponential function $f(x) = b^x$. The primary constraint $b > 0$ is imposed to ensure the function is defined and continuous over the entire real domain, $x \in \mathbb{R}$. If we attempted to define the function with a negative base, such as $b = -9$, for which of the following values of $x$ would $f(x)$ *not* result in a real number?

    $x = 2$

    $x = 0$

    $x = \frac{1}{2}$

    $x = -1$

17. The formal definition of an exponential function $f(x) = b^x$ requires the base $b$ to satisfy $b \neq 1$. If $b$ were equal to $1$, the resulting function $g(x) = 1^x$ would simplify to $g(x)=1$. This constraint is necessary primarily because:

    The function $g(x)$ is a linear function with a slope of zero, violating non-linear growth.

    The function $g(x)=1$ is a constant function, lacking the defining characteristic of rapid, non-linear growth or decay necessary for the formal exponential classification.

    The domain of $g(x)$ is restricted only to positive integers.

    The range of $g(x)$ is $\{1\}$, which violates the asymptotic behavior required of exponential graphs.

18. State the complete necessary and sufficient constraints on the base $b$ for the standard formal exponential function $f(x) = b^x$ using proper interval notation.

    $(0, 1) \cup (1, \infty)$

    $(-\infty, 0) \cup (0, \infty)$

    $(0, \infty)$

    $[0, 1) \cup (1, \infty)$

19. For a formal definition of $f(x)=b^x$ to be complete for all real numbers $x$, the values for irrational exponents (e.g., $b^{\pi}$) must be defined. This definition relies on approaching the irrational exponent via a sequence of rational exponents. Which mathematical property ensures that the value of $b^x$ is unique and consistent across the domain, allowing this transition from rational to irrational exponents?

    The base $b$ must be a rational number or $e$.

    The domain must be restricted to non-negative real numbers.

    The function must exhibit strict monotonicity (always increasing or always decreasing).

    The function must be continuous over the domain $\mathbb{R}$.

20. A defining consequence of the formal constraint $b>0$ in $f(x)=b^x$ is that the range of the standard exponential function is always $R: (0, \infty)$, meaning all outputs are strictly positive. If we relax the constraint and allow a negative base $b < 0$, but restrict the domain to only integer values $x \in \mathbb{Z}$ to ensure real outputs, how would the range fundamentally differ from the standard range $R$?

    The outputs would alternate between positive and negative values, thus violating the non-negativity constraint inherent in the standard exponential range.

    The range would include $y=0$ as an asymptote.

    The range would consist only of rational numbers.

    The function would require a vertical shift to maintain strict positivity.

21. For an exponential function defined by $f(x) = a \cdot b^x$, where $a > 0$, the base $b$ dictates whether the function exhibits growth or decay. Which pair of conditions correctly defines $b$ for growth and decay, respectively, while ensuring the function is truly exponential?

    Growth: $b \ge 1$; Decay: $0 < b < 1$

    Growth: $b > 0$; Decay: $b < 0$

    Growth: $b > 1$; Decay: $0 < b < 1$

    Growth: $b$ is any real number greater than 1; Decay: $b$ is any real number between $-1$ and $1$

22. A local conservation group models the decline of a fish population using the formula $P(t) = P_0 (1 - r)^t$, where $r$ is the annual rate of decrease. If the population decreases by $12.5\%$ annually, what is the exact value of the base $b$ in the equivalent form $P(t) = P_0 (b)^t$, and what type of behavior does it represent?

    $b = 1.125$, representing exponential growth.

    $b = 1 - \frac{1}{8}$, representing exponential growth.

    $b = 0.875$, representing exponential decay.

    $b = 0.125$, representing exponential decay.

23. Consider the function $f(x) = C \cdot b^x$. If $b$ is defined as $b = \frac{1}{\sqrt{2} + 1}$, does $f(x)$ represent exponential growth or exponential decay? (Hint: $\sqrt{2} \approx 1.414$)

    Exponential Growth, because the denominator is greater than the numerator.

    Exponential Growth, because $b$ is an irrational number.

    Cannot be determined unless the value of $C$ is known.

    Exponential Decay, because $b$ is approximately $0.414$, which is less than 1.

24. An investor compares two savings models starting with the same principal $P$: $M_1(t) = P(1.05)^t$ and $M_2(t) = P(0.95)^t$. Which statement accurately compares the instantaneous rate of change (steepness of the curve) of these models near $t=0$?

    $M_2(t)$ must have a steeper rate of change because decay functions decrease rapidly.

    The magnitude of the rate of change is nearly identical near $t=0$, because both bases are equally distant from the constant boundary $b=1$.

    $M_1(t)$ is much steeper than $M_2(t)$ because the base for growth is larger than the base for decay.

    Since $1.05$ and $0.95$ are both positive, their rates of change are identical across the entire domain.

25. For an equation structured as $y = a \cdot b^x$, which specific condition on the base $b$ causes the function to exhibit neither exponential growth nor exponential decay, resulting instead in a constant output regardless of the input $x$?

    $b$ is between $-1$ and $0$

    $b = 1$

    $b = 0$

    $b$ is negative, such as $b = -2$

26. Solve for $x$: $2^x = 8$.

    $8$

    $3$

    $2$

    $16$

27. Solve for $x$: $3^x = 81$.

    $4$

    $27$

    $3$

    $9$

28. Determine the value of $x$ that satisfies $4^{x+1} = 16$.

    $3$

    $4$

    $2$

    $1$

29. Solve the equation $5^x = \frac{1}{25}$.

    $-5$

    $-2$

    $2$

    $\frac{1}{2}$

30. What is the solution to $2^x = \frac{1}{16}$?

    $4$

    $-8$

    $-2$

    $-4$

31. Find $x$: $3^{2x} = \frac{1}{9}$.

    $1$

    $-1$

    $2$

    $-2$

32. Solve for $x$: $4^{x} = 32$.

    $\frac{5}{2}$

    $2$

    $\frac{5}{4}$

    $5$

33. If $9^{x} = 27$, what is $x$?

    $\frac{3}{4}$

    $\frac{3}{2}$

    $1$

    $3$

34. Solve for $x$: $10^{2x-1} = 1000$.

    $2$

    $3$

    $1$

    $0$

35. Find the value of $x$ in the equation $5^{x-3} = 125$.

    $3$

    $4$

    $6$

    $0$

36. Given $2^{3x+1} = 4$, determine $x$.

    $-1$

    $3$

    $\frac{1}{3}$

    $1$

37. Solve: $7^{4x} = \frac{1}{49}$.

    $1$

    $-\frac{1}{2}$

    $-\frac{1}{4}$

    $2$

38. Solve for $x$: $8^{x} = \frac{1}{4}$.

    $2$

    $\frac{2}{3}$

    $-\frac{3}{2}$

    $-\frac{2}{3}$

39. Find $x$ such that $27^{x-1} = 9^{2x}$.

    $\frac{3}{7}$

    $3$

    $-3$

    $-1$

40. Solve $6^{x+5} = 36^{x}$.

    $-5$

    $5$

    $10$

    $2$

41. Solve for $x$: $27^{x-1} = \frac{1}{9^x}$.

    $x = -3$

    $x = 3/5$

    $x = 3/7$

    $x = 3$

42. Find the positive solution for $x$: $4^{x^2 - 4} = 8^{x+2}$.

    $x = 2$

    $x = 4$

    $x = -2$

    $x = 7/2$

43. Solve for $x$: $125^{2x} = \sqrt{5^{x+1}}$.

    $x = 1/6$

    $x = 1/11$

    $x = 1/7$

    $x = 1/13$

44. Determine the value of $x$ that satisfies $16^{x+3} = 4 \cdot 8^{2x}$.

    $x = 1$

    $x = 5$

    $x = 4$

    $x = -2$

45. Find the solution to $(\frac{1}{8})^{2x-3} = 32^{x+1}$.

    $x = 1$

    $x = 4/5$

    $x = 14/11$

    $x = 4/11$

46. If $3^{x^2} = 9^{x+4}$, determine the negative value of $x$.

    $x = -4$

    $x = 4$

    $x = -1$

    $x = -2$

47. Solve for the non-zero value of $x$: $100^{x^2} = 1000^x$.

    $x = 1/2$

    $x = 3$

    $x = 2/3$

    $x = 3/2$

48. If $\frac{1}{64} = (\sqrt[3]{4})^{x+1}$, what is the value of $x$?

    $x = -10$

    $x = -12$

    $x = 8$

    $x = -9$

49. Solve for $x$: $\frac{25^{3x+1}}{5^{x-2}} = 125$.

    $x = 3/5$

    $x = -1$

    $x = -5$

    $x = -1/5$

50. Find $x$ if $6 \cdot 36^{x-1} = \frac{1}{6^{2x}}$.

    $x = 1/2$

    $x = -1/4$

    $x = 3/2$

    $x = 1/4$