Algebra Flashcards: Rewrite Exponential Expressions Using Exponents

What this deck covers

This deck focuses on Rewrite Exponential Expressions Using Exponents, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 51: Rewrite $\frac{10^{3t}}{10^{t}}$ as a base raised to $t$.

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

Flashcard 52: Rewrite $\left(2^t\right)^5$ as a single exponential expression.

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

Flashcard 53: State the exponent rule that rewrites $a^m a^n$ as a single power.

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

Flashcard 54: Find and correct the error: $\left(3^4\right)^t = 3^{4+t}$.

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

Flashcard 55: Rewrite $\frac{5^{2t}}{5^t}$ as a single exponential expression.

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