Algebra 2 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 2.

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

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

Flashcard 2: Identify the base after rewriting $b^t$ as $\left(b^{\frac{1}{n}}\right)^{nt}$.

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

Flashcard 3: Rewrite $b^{2t}$ to show $6t$ in the exponent: $b^{2t} = (?)^{6t}$.

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

Flashcard 4: Identify the exponent property used to justify $b^t=\left(b^{\frac{1}{n}}\right)^{nt}$.

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

Flashcard 5: State the exponent rule that rewrites $a^{mn}$ using a power raised to a power.

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

Flashcard 6: State the exponent rule that rewrites $(a^m)^n$ as a single power of $a$.

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

Flashcard 7: State the rule for multiplying same-base powers: $a^m\cdot a^n$.

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

Flashcard 8: State the rule for dividing same-base powers: $\frac{a^m}{a^n}$ for $a\ne 0$.

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

Flashcard 9: State the meaning of a zero exponent for $a\ne 0$.

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

Flashcard 10: State the meaning of a negative exponent for $a\ne 0$.

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

Flashcard 11: State the meaning of a fractional exponent $a^{\frac{1}{n}}$ for $a\ge 0$.

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

Flashcard 12: State the meaning of $a^{\frac{m}{n}}$ for $a\ge 0$ and integers $m,n>0$.

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

Flashcard 13: What is the standard form for an exponential function with initial value $a$ and factor $b$?

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

Flashcard 14: What base $b$ corresponds to exponential growth in $ab^t$?

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

Flashcard 15: What base $b$ corresponds to exponential decay in $ab^t$?

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

Flashcard 16: State the transformation that converts $b^t$ into a form with $n$ periods per unit: $b^t=(\,?\,)^{nt}$.

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

Flashcard 17: Identify the per-period growth factor if the annual factor is $b$ and there are $n$ compounding periods per year.

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

Flashcard 18: Identify the per-period rate $r$ if the per-period factor is $k$.

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

Flashcard 19: Identify the per-period factor $k$ if the per-period rate is $r$.

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

Flashcard 20: What is the monthly factor equivalent to an annual factor of $1.15$ (symbolic form only)?

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

Flashcard 21: Rewrite $1.15^t$ to show $12$ equal growth periods per year using exponent properties.

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

Flashcard 22: Rewrite $b^{3t}$ as a power with exponent $t$ (single-step rewrite).

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

Flashcard 23: Rewrite $(b^3)^t$ as a single exponential expression in the form $b^{\,?\,}$.

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

Flashcard 24: Rewrite $b^{\frac{t}{5}}$ in the form $(\,?\,)^{t}$ using exponent properties.

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

Flashcard 25: Rewrite $(b^{\frac{1}{5}})^t$ as a single power of $b$.

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

Flashcard 26: Rewrite $b^{t+7}$ as a product involving $b^t$ using exponent properties.

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

Flashcard 27: Rewrite $b^{t-4}$ as a quotient involving $b^t$ using exponent properties.

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

Flashcard 28: Rewrite $\frac{b^{t}}{b^{2t}}$ as a single power of $b$.

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

Flashcard 29: Rewrite $b^{-t}$ without negative exponents.

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

Flashcard 30: Find the equivalent expression for $\left(1+r\right)^{12t}$ written as $\left(\,?\,\right)^t$.

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