Rewrite Exponential Expressions Using Exponents - Algebra 1

1. Rewrite $8^{\frac{t}{3}}$ as a power with exponent $t$.

   $8^{\frac{t}{3}} = 2^t$. Since $8 = 2^3$, we get $(2^3)^{t/3} = 2^t$.

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

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

3. Rewrite $1.21^t$ to show a quarterly factor raised to $4t$.

   $1.21^t = \left(1.21^{\frac{1}{4}}\right)^{4t}$. Rewrites annual factor as quarterly factor raised to 4 times the power.

4. Rewrite $2^{6t}$ as a base raised to $t$.

   $2^{6t} = 64^t$. Since $2^6 = 64$, using power of a power rule gives $64^t$.

5. State the exponent rule that rewrites $\frac{a^m}{a^n}$ as a single power (for $a \neq 0$).

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

6. Identify the annual factor $a$ if the annual interest rate is $R$ (as a decimal).

   $a = 1 + R$. Growth factor is 1 plus the interest rate.

7. Identify the equivalent weekly factor if the annual factor is $0.80$.

   $0.80^{\frac{1}{52}}$. Weekly factor is the 52nd root of the annual factor.

8. Rewrite $\frac{10^{3t}}{10^{t}}$ as a base raised to $t$.

   $\frac{10^{3t}}{10^{t}} = 10^{2t} = 100^t$. Quotient rule gives $10^{2t}$, then $(10^2)^t = 100^t$.

9. Rewrite $\left(1.5^2\right)^t$ as a single exponential expression with base $1.5$.

   $\left(1.5^2\right)^t = 1.5^{2t}$. Power of a power rule: $(1.5^2)^t = 1.5^{2t}$.

10. Find and correct the error: $\frac{5^{2t}}{5^t} = 5^{2t}$.

    Correct: $\frac{5^{2t}}{5^t} = 5^t$. Quotient rule subtracts exponents: $2t - t = t$, not $2t$.

11. Rewrite $\frac{2^{t+1}}{2^{t-2}}$ as a constant (no variable exponent).

    $\frac{2^{t+1}}{2^{t-2}} = 2^3 = 8$. Quotient rule: $\frac{2^{t+1}}{2^{t-2}} = 2^{(t+1)-(t-2)} = 2^3 = 8$.

12. State the exponent rule for rewriting $\left(\frac{a}{b}\right)^n$ for $b \neq 0$.

    $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. Quotient to power rule: distribute the exponent to numerator and denominator.

13. What transformation rewrites $b^t$ as an equivalent expression with exponent $kt$?

    $b^t = \left(b^{\frac{1}{k}}\right)^{kt}$. Uses power of a power rule: $(b^{1/k})^{kt} = b^{(1/k) \cdot kt} = b^t$.

14. Find and correct the error: $\left(1.10^t\right)^{12} = 1.10^{t+12}$.

    Correct: $\left(1.10^t\right)^{12} = 1.10^{12t}$. Power of a power rule multiplies exponents, not adds them.

15. Rewrite $27^{\frac{t}{3}}$ as a base raised to $t$.

    $27^{\frac{t}{3}} = 3^t$. Since $27 = 3^3$, we get $(3^3)^{t/3} = 3^t$.

16. What is the monthly factor $m$ for a $15\%$ annual rate, written exactly using exponents?

    $m = 1.15^{\frac{1}{12}}$. 15% annually means factor 1.15, so monthly is $1.15^{1/12}$.

17. Identify the monthly interest rate $r$ if the monthly factor is $m$.

    $r = m - 1$. Interest rate is the growth factor minus 1.

18. Identify the daily growth factor $d$ if the annual factor is $a$ and there are $365$ days.

    $d = a^{\frac{1}{365}}$. Daily factor is the 365th root of the annual factor.

19. Rewrite $\left(7^t\right)\left(7^{-t}\right)$ as a constant.

    $\left(7^t\right)\left(7^{-t}\right) = 7^0 = 1$. Product rule: $7^t \cdot 7^{-t} = 7^{t+(-t)} = 7^0 = 1$.

20. Rewrite $a^{\frac{t}{n}}$ as a power with exponent $t$.

    $a^{\frac{t}{n}} = \left(a^{\frac{1}{n}}\right)^t$. Uses power of a power rule to rewrite with exponent $t$.

21. Rewrite $1.15^t$ in the form $\left(1.15^{\frac{1}{12}}\right)^{12t}$.

    $1.15^t = \left(1.15^{\frac{1}{12}}\right)^{12t}$. Applies the power of a power rule: $(a^{1/n})^{nt} = a^t$.

22. What is the general formula for the periodic factor if $a$ is the factor per 1 time unit and there are $n$ periods?

    $\text{period factor} = a^{\frac{1}{n}}$. Periodic factor is the nth root of the factor per unit time.

23. Rewrite $\left(0.99\right)^{365t}$ as an equivalent annual factor raised to $t$.

    $\left(0.99\right)^{365t} = \left(0.99^{365}\right)^t$. Power of a power rule: $(0.99^{365})^t = 0.99^{365t}$.

24. Rewrite $4^{3t}$ as a power with exponent $t$.

    $4^{3t} = 64^t$. Since $4^3 = 64$, using power of a power rule gives $(4^3)^t = 64^t$.

25. State the exponent rule for rewriting $(ab)^n$.

    $(ab)^n = a^n b^n$. Product to power rule: distribute the exponent to each factor.

26. Rewrite $\left(\frac{9}{4}\right)^{\frac{t}{2}}$ as a base raised to $t$.

    $\left(\frac{9}{4}\right)^{\frac{t}{2}} = \left(\frac{3}{2}\right)^t$. Since $9/4 = (3/2)^2$, we get $((3/2)^2)^{t/2} = (3/2)^t$.

27. Rewrite $9^{\frac{t}{2}}$ as a base raised to $t$.

    $9^{\frac{t}{2}} = 3^t$. Since $9 = 3^2$, we get $(3^2)^{t/2} = 3^t$.

28. What is the definition of a zero exponent for $a \neq 0$?

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

29. Rewrite $16^{\frac{t}{4}}$ as a base raised to $t$.

    $16^{\frac{t}{4}} = 2^t$. Since $16 = 2^4$, we get $(2^4)^{t/4} = 2^t$.

30. Rewrite $\left(\left(3^t\right)^2\right)^4$ as a single exponential expression.

    $\left(\left(3^t\right)^2\right)^4 = 3^{8t}$. Applying power rules repeatedly: $((3^t)^2)^4 = (3^{2t})^4 = 3^{8t}$.

31. Identify the monthly growth factor $m$ if the annual factor is $a$ and there are $12$ months.

    $m = a^{\frac{1}{12}}$. Monthly factor is the 12th root of the annual factor.

32. Rewrite $\left(\frac{16}{25}\right)^{\frac{t}{2}}$ as a base raised to $t$.

    $\left(\frac{16}{25}\right)^{\frac{t}{2}} = \left(\frac{4}{5}\right)^t$. Since $16/25 = (4/5)^2$, we get $((4/5)^2)^{t/2} = (4/5)^t$.

33. Rewrite $\left(\frac{1}{4}\right)^{\frac{t}{2}}$ as a base raised to $t$.

    $\left(\frac{1}{4}\right)^{\frac{t}{2}} = \left(\frac{1}{2}\right)^t$. Since $1/4 = (1/2)^2$, we get $((1/2)^2)^{t/2} = (1/2)^t$.

34. Rewrite $\left(1.02\right)^{12t}$ as an equivalent annual factor raised to $t$.

    $\left(1.02\right)^{12t} = \left(1.02^{12}\right)^t$. Power of a power rule: $(1.02^{12})^t = 1.02^{12t}$.

35. Rewrite $\left(\frac{1}{3}\right)^{2t}$ as a base raised to $t$.

    $\left(\frac{1}{3}\right)^{2t} = \left(\frac{1}{9}\right)^t$. Since $(1/3)^2 = 1/9$, using power rule gives $(1/9)^t$.

36. Identify the equivalent monthly factor if the annual factor is $1.06$.

    $1.06^{\frac{1}{12}}$. Monthly factor is the 12th root of the annual factor.

37. What is the definition of a negative exponent for $a \neq 0$?

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

38. Rewrite $81^{\frac{t}{4}}$ as a base raised to $t$.

    $81^{\frac{t}{4}} = 3^t$. Since $81 = 3^4$, we get $(3^4)^{t/4} = 3^t$.

39. Rewrite $\left(1.01\right)^{30t}$ as an equivalent factor raised to $t$.

    $\left(1.01\right)^{30t} = \left(1.01^{30}\right)^t$. Power of a power rule: $(1.01^{30})^t = 1.01^{30t}$.

40. Rewrite $\left(2^t\right)\left(2^{3t}\right)$ as a single exponential expression.

    $\left(2^t\right)\left(2^{3t}\right) = 2^{4t}$. Product rule: $2^t \cdot 2^{3t} = 2^{t+3t} = 2^{4t}$.

41. Identify the weekly growth factor $w$ if the annual factor is $a$ and there are $52$ weeks.

    $w = a^{\frac{1}{52}}$. Weekly factor is the 52nd root of the annual factor.

42. Rewrite $5^{t+3}$ as a product involving $5^t$.

    $5^{t+3} = 125\cdot 5^t$. Using $a^{m+n} = a^m \cdot a^n$, so $5^{t+3} = 5^3 \cdot 5^t = 125 \cdot 5^t$.

43. What transformation rewrites $b^{kt}$ as an equivalent expression with exponent $t$?

    $b^{kt} = \left(b^k\right)^t$. Uses power of a power rule: $(b^k)^t = b^{kt}$.

44. Identify the equivalent quarterly factor if the annual factor is $1.12$.

    $1.12^{\frac{1}{4}}$. Quarterly factor is the 4th root of the annual factor.

45. Rewrite $1.08^t$ to show a monthly factor raised to $12t$.

    $1.08^t = \left(1.08^{\frac{1}{12}}\right)^{12t}$. Rewrites annual factor as monthly factor raised to 12 times the power.

46. Rewrite $1.44^t$ to show a monthly factor raised to $12t$.

    $1.44^t = \left(1.44^{\frac{1}{12}}\right)^{12t}$. Rewrites annual factor as monthly factor raised to 12 times the power.

47. Rewrite $10^{t-2}$ as a product involving $10^t$.

    $10^{t-2} = \frac{1}{100}\cdot 10^t$. Using $a^{m-n} = \frac{a^m}{a^n}$, so $10^{t-2} = \frac{10^t}{10^2} = \frac{1}{100} \cdot 10^t$.

48. Rewrite $\left(b^t\right)^3$ as a single exponential expression in $b$ and $t$.

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

49. Rewrite $32^{\frac{t}{5}}$ as a power with exponent $t$.

    $32^{\frac{t}{5}} = 2^t$. Since $32 = 2^5$, we get $(2^5)^{t/5} = 2^t$.

50. Rewrite $3^{2t}$ as a power with exponent $t$.

    $3^{2t} = 9^t$. Since $3^2 = 9$, using power of a power rule gives $(3^2)^t = 9^t$.

51. Rewrite $\left(2^t\right)^5$ as a single exponential expression.

    $\left(2^t\right)^5 = 2^{5t}$. Power of a power rule: $(2^t)^5 = 2^{5t}$.

52. State the exponent rule that rewrites $a^m a^n$ as a single power.

    $a^m a^n = a^{m+n}$. Product of powers rule: add the exponents when bases are equal.

53. Find and correct the error: $\left(3^4\right)^t = 3^{4+t}$.

    Correct: $\left(3^4\right)^t = 3^{4t}$. Power of a power rule multiplies exponents, not adds them.

54. Rewrite $\frac{5^{2t}}{5^t}$ as a single exponential expression.

    $\frac{5^{2t}}{5^t} = 5^t$. Quotient rule: $\frac{5^{2t}}{5^t} = 5^{2t-t} = 5^t$.

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