Rewrite Exponential Expressions Using Exponents - Algebra 2

1. What is the standard form for an exponential function with initial value $a$ and factor $b$?

   $f(t)=ab^t$. Standard exponential form with base $b$ and coefficient $a$.

2. Identify the per-period rate $r$ if the per-period factor is $k$.

   $r=k-1$. Rate is factor minus 1.

3. Rewrite $b^{t-4}$ as a quotient involving $b^t$ using exponent properties.

   $b^{t-4}=\frac{b^t}{b^4}$. Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$.

4. What base $b$ corresponds to exponential growth in $ab^t$?

   $b>1$. Base greater than 1 causes exponential growth.

5. What is the weekly factor equivalent to an annual factor of $b$ assuming $52$ weeks per year?

   $b^{\frac{1}{52}}$. 52nd root of annual factor gives weekly factor.

6. Identify the annual factor if the monthly rate is $p$ (as a decimal) so monthly factor is $1+p$.

   $(1+p)^{12}$. 12th power of monthly factor $(1+p)$.

7. State the meaning of $a^{\frac{m}{n}}$ for $a\ge 0$ and integers $m,n>0$.

   $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. Fractional exponent: raise to m, then take nth root.

8. Rewrite $\left(1.08\right)^{t}$ to show a monthly factor with exponent $12t$ (symbolic only).

   $\left(1.08^{\frac{1}{12}}\right)^{12t}$. Monthly factor with 12 periods per year.

9. Rewrite $\left(0.90\right)^{t}$ to show a monthly decay factor with exponent $12t$ (symbolic only).

   $\left(0.90^{\frac{1}{12}}\right)^{12t}$. Monthly decay factor with 12 periods per year.

10. Find and correct the exponent error: $\left(b^{\frac{1}{12}}\right)^{t}=b^{12t}$.

    Correct: $\left(b^{\frac{1}{12}}\right)^t=b^{\frac{t}{12}}$. Power of a power: multiply exponents $\frac{1}{12} \times t$.

11. Identify the base after rewriting $b^t$ as $\left(b^{\frac{1}{n}}\right)^{nt}$.

    $b^{\frac{1}{n}}$. The new base after transformation.

12. State the exponent rule that rewrites $a^{mn}$ using a power raised to a power.

    $a^{mn}=(a^m)^n$. Power of a power rule: multiply the exponents.

13. Identify the per-period growth factor if the annual factor is $b$ and there are $n$ compounding periods per year.

    $b^{\frac{1}{n}}$. nth root of annual factor gives per-period factor.

14. What is the monthly factor equivalent to an annual factor of $1.15$ (symbolic form only)?

    $1.15^{\frac{1}{12}}$. 12th root of annual factor gives monthly factor.

15. Rewrite $b^{t+7}$ as a product involving $b^t$ using exponent properties.

    $b^{t+7}=b^t\cdot b^7$. Product rule: $a^m \cdot a^n = a^{m+n}$.

16. Find the equivalent expression for $\left(1+r\right)^{12t}$ written as $\left(\,?\,\right)^t$.

    $\left(1+r\right)^{12t}=\left(\left(1+r\right)^{12}\right)^t$. Power of a power rule applied in reverse.

17. State the meaning of a zero exponent for $a\ne 0$.

    $a^0=1$. Any nonzero number to the zero power equals 1.

18. State the meaning of a negative exponent for $a\ne 0$.

    $a^{-n}=\frac{1}{a^n}$. Negative exponent means reciprocal of positive power.

19. Rewrite $a\left(1+r\right)^{t}$ to show $n$ compounding periods per unit time.

    $a\left(\left(1+r\right)^{\frac{1}{n}}\right)^{nt}$. General form for $n$ compounding periods.

20. Rewrite $ab^{t}$ to show $n$ equal periods per unit: $a( \, ? \, )^{nt}$.

    $a\left(b^{\frac{1}{n}}\right)^{nt}$. General form for $n$ periods per unit time.

21. What base $b$ corresponds to exponential decay in $ab^t$?

    $0<b<1$. Base between 0 and 1 causes exponential decay.

22. Rewrite $\left(\sqrt[5]{b}\right)^{t}$ using rational exponents.

    $b^{\frac{t}{5}}$. Fractional exponent $\frac{1}{5}$ means 5th root.

23. Rewrite $b^{3t}$ as a power with exponent $t$ (single-step rewrite).

    $b^{3t}=(b^3)^t$. Power of a power rule: $(a^m)^n = a^{mn}$.

24. Find the per-period factor if $\left(1+r\right)^{t}=k^{nt}$ for $n$ equal periods per unit.

    $k=\left(1+r\right)^{\frac{1}{n}}$. Per-period factor is nth root of annual factor.

25. Rewrite $b^{2t}$ to show $6t$ in the exponent: $b^{2t} = (?)^{6t}$.

    $\left(b^{\frac{1}{3}}\right)^{6t}$. Rewrite using fractional exponent of $\frac{1}{3}$.

26. Rewrite $b^{t}$ as $\left(\sqrt{b}\right)^{\,?\,}$ to show twice as many periods.

    $\left(\sqrt{b}\right)^{2t}$. Square root creates twice as many periods.

27. Find and correct the exponent error: $b^t=\left(b^{12}\right)^{\frac{t}{12}}$.

    Correct: $b^t=\left(b^{\frac{1}{12}}\right)^{12t}$. Power of a power rule requires fractional base exponent.

28. Rewrite $1.15^t$ to show $12$ equal growth periods per year using exponent properties.

    $1.15^t=\left(1.15^{\frac{1}{12}}\right)^{12t}$. Power of a power rule creates monthly compounding.

29. Rewrite $\left(\sqrt{b}\right)^{2t}$ as a single power of $b$.

    $b^{t}$. $(\sqrt{b})^{2t} = (b^{\frac{1}{2}})^{2t} = b^t$.

30. Rewrite $(b^3)^t$ as a single exponential expression in the form $b^{\,?\,}$.

    $b^{3t}$. Power of a power rule: $(a^m)^n = a^{mn}$.

31. State the meaning of a fractional exponent $a^{\frac{1}{n}}$ for $a\ge 0$.

    $a^{\frac{1}{n}}=\sqrt[n]{a}$. Fractional exponent means nth root.

32. Rewrite $b^{\frac{t}{5}}$ in the form $( ? )^{t}$ using exponent properties.

    $\left(b^{\frac{1}{5}}\right)^t$. Fractional exponent becomes root in the base.

33. State the rule for multiplying same-base powers: $a^m\cdot a^n$.

    $a^m\cdot a^n=a^{m+n}$. Product of powers rule: add the exponents.

34. Rewrite $(b^{\frac{1}{5}})^t$ as a single power of $b$.

    $b^{\frac{t}{5}}$. Power of a power rule: $(a^m)^n = a^{mn}$.

35. Rewrite $\left(1.21\right)^{t}$ to show a quarterly factor with exponent $4t$ (symbolic only).

    $\left(1.21^{\frac{1}{4}}\right)^{4t}$. Quarterly factor with 4 periods per year.

36. State the rule for dividing same-base powers: $\frac{a^m}{a^n}$ for $a \neq 0$.

    $\frac{a^m}{a^n}=a^{m-n}$. Quotient of powers rule: subtract the exponents.

37. Rewrite $ab^{t}$ to show $n$ equal periods per unit: $a(\,?\,)^{nt}$.

    $a\left(b^{\frac{1}{n}}\right)^{nt}$. General form for $n$ periods per unit time.

38. Rewrite $\frac{b^{t}}{b^{2t}}$ as a single power of $b$.

    $b^{-t}$. Quotient rule: $\frac{b^t}{b^{2t}} = b^{t-2t} = b^{-t}$.

39. Rewrite $\left(b^{\frac{1}{3}}\right)^{6t}$ as a single power of $b$.

    $b^{2t}$. Power of a power: $(b^{\frac{1}{3}})^{6t} = b^{\frac{6t}{3}} = b^{2t}$.

40. State the transformation that converts $b^t$ into a form with $n$ periods per unit: $b^t=(\,?\,)^{nt}$.

    $b^t=\left(b^{\frac{1}{n}}\right)^{nt}$. Power of a power rule applied to create $n$ periods.

41. Rewrite $a\left(b^{\frac{1}{12}}\right)^{12t}$ in the form $ab^t$ (single-step).

    $ab^t$. Power of a power rule: $(a^m)^n = a^{mn}$.

42. Rewrite $b^{\frac{t}{5}}$ in the form $(\,?\,)^{t}$ using exponent properties.

    $\left(b^{\frac{1}{5}}\right)^t$. Fractional exponent becomes root in the base.

43. Identify the annual factor if the monthly factor is $m$ (compounded monthly).

    $m^{12}$. 12 monthly factors multiply to give annual factor.

44. State the rule for dividing same-base powers: $\frac{a^m}{a^n}$ for $a\ne 0$.

    $\frac{a^m}{a^n}=a^{m-n}$. Quotient of powers rule: subtract the exponents.

45. Rewrite $b^{\frac{t}{8}}$ using a radical expression raised to $t$.

    $\left(\sqrt[8]{b}\right)^t$. Fractional exponent $\frac{1}{8}$ means 8th root.

46. Identify the per-period factor $k$ if the per-period rate is $r$.

    $k=1+r$. Factor is 1 plus the rate.

47. Rewrite $b^t$ to show $2$ equal periods per unit time (semiannual) using exponent properties.

    $b^t=\left(b^{\frac{1}{2}}\right)^{2t}$. Power of a power rule creates semiannual compounding.

48. Rewrite $\left(\left(1+r\right)^{12}\right)^t$ as a single exponent on $1+r$.

    $\left(1+r\right)^{12t}$. Power of a power rule: $ $(a^m)^n$ = $a^{mn}$ $.

49. Identify the monthly factor if the annual percentage rate is $r$ (as a decimal) with annual factor $1+r$.

    $(1+r)^{\frac{1}{12}}$. 12th root of annual factor $(1+r)$.

50. Identify the exponent property used to justify $b^t=\left(b^{\frac{1}{n}}\right)^{nt}$.

    $(a^m)^n=a^{mn}$. This property allows rewriting for multiple periods.

51. Rewrite $b^{-t}$ without negative exponents.

    $\frac{1}{b^t}$. Negative exponent becomes reciprocal.

52. Rewrite $b^t$ to show $4$ equal periods per unit time (quarterly) using exponent properties.

    $b^t=\left(b^{\frac{1}{4}}\right)^{4t}$. Power of a power rule creates quarterly compounding.

53. Identify the daily factor if the annual factor is $b$ and there are $365$ equal periods per year.

    $b^{\frac{1}{365}}$. 365th root of annual factor gives daily factor.

54. Identify the new exponent after rewriting $b^t$ as $\left(b^{\frac{1}{n}}\right)^{nt}$.

    $nt$. The new exponent after transformation.

55. State the exponent rule that rewrites $(a^m)^n$ as a single power of $a$.

    $(a^m)^n=a^{mn}$. Power of a power rule: multiply the exponents.

56. Find the monthly rate (symbolic) equivalent to an annual rate of $15\%$ with factor $1.15$.

    $1.15^{\frac{1}{12}}-1$. Monthly rate is monthly factor minus 1.

Study these flashcards