Rewrite Exponential Expressions Using Exponents - Algebra
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Rewrite $\left(b^t\right)^3$ as a single exponential expression in $b$ and $t$.
Rewrite $\left(b^t\right)^3$ as a single exponential expression in $b$ and $t$.
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$\left(b^t\right)^3 = b^{3t}$. Power of a power rule: $(a^m)^n = a^{mn}$.
$\left(b^t\right)^3 = b^{3t}$. Power of a power rule: $(a^m)^n = a^{mn}$.
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Find and correct the error: $\frac{5^{2t}}{5^t} = 5^{2t}$.
Find and correct the error: $\frac{5^{2t}}{5^t} = 5^{2t}$.
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Correct: $\frac{5^{2t}}{5^t} = 5^t$. Quotient rule subtracts exponents: $2t - t = t$, not $2t$.
Correct: $\frac{5^{2t}}{5^t} = 5^t$. Quotient rule subtracts exponents: $2t - t = t$, not $2t$.
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Find and correct the error: $\left(3^4\right)^t = 3^{4+t}$.
Find and correct the error: $\left(3^4\right)^t = 3^{4+t}$.
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Correct: $\left(3^4\right)^t = 3^{4t}$. Power of a power rule multiplies exponents, not adds them.
Correct: $\left(3^4\right)^t = 3^{4t}$. Power of a power rule multiplies exponents, not adds them.
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Rewrite $\left(2^t\right)^5$ as a single exponential expression.
Rewrite $\left(2^t\right)^5$ as a single exponential expression.
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$\left(2^t\right)^5 = 2^{5t}$. Power of a power rule: $(2^t)^5 = 2^{5t}$.
$\left(2^t\right)^5 = 2^{5t}$. Power of a power rule: $(2^t)^5 = 2^{5t}$.
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Rewrite $\left(\left(3^t\right)^2\right)^4$ as a single exponential expression.
Rewrite $\left(\left(3^t\right)^2\right)^4$ as a single exponential expression.
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$\left(\left(3^t\right)^2\right)^4 = 3^{8t}$. Applying power rules repeatedly: $((3^t)^2)^4 = (3^{2t})^4 = 3^{8t}$.
$\left(\left(3^t\right)^2\right)^4 = 3^{8t}$. Applying power rules repeatedly: $((3^t)^2)^4 = (3^{2t})^4 = 3^{8t}$.
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Rewrite $\frac{2^{t+1}}{2^{t-2}}$ as a constant (no variable exponent).
Rewrite $\frac{2^{t+1}}{2^{t-2}}$ as a constant (no variable exponent).
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$\frac{2^{t+1}}{2^{t-2}} = 2^3 = 8$. Quotient rule: $\frac{2^{t+1}}{2^{t-2}} = 2^{(t+1)-(t-2)} = 2^3 = 8$.
$\frac{2^{t+1}}{2^{t-2}} = 2^3 = 8$. Quotient rule: $\frac{2^{t+1}}{2^{t-2}} = 2^{(t+1)-(t-2)} = 2^3 = 8$.
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Rewrite $\frac{5^{2t}}{5^t}$ as a single exponential expression.
Rewrite $\frac{5^{2t}}{5^t}$ as a single exponential expression.
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$\frac{5^{2t}}{5^t} = 5^t$. Quotient rule: $\frac{5^{2t}}{5^t} = 5^{2t-t} = 5^t$.
$\frac{5^{2t}}{5^t} = 5^t$. Quotient rule: $\frac{5^{2t}}{5^t} = 5^{2t-t} = 5^t$.
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Rewrite $a^{\frac{t}{n}}$ as a power with exponent $t$.
Rewrite $a^{\frac{t}{n}}$ as a power with exponent $t$.
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$a^{\frac{t}{n}} = \left(a^{\frac{1}{n}}\right)^t$. Uses power of a power rule to rewrite with exponent $t$.
$a^{\frac{t}{n}} = \left(a^{\frac{1}{n}}\right)^t$. Uses power of a power rule to rewrite with exponent $t$.
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Rewrite $\frac{10^{3t}}{10^{t}}$ as a base raised to $t$.
Rewrite $\frac{10^{3t}}{10^{t}}$ as a base raised to $t$.
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$\frac{10^{3t}}{10^{t}} = 10^{2t} = 100^t$. Quotient rule gives $10^{2t}$, then $(10^2)^t = 100^t$.
$\frac{10^{3t}}{10^{t}} = 10^{2t} = 100^t$. Quotient rule gives $10^{2t}$, then $(10^2)^t = 100^t$.
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Rewrite $\left(2^t\right)\left(2^{3t}\right)$ as a single exponential expression.
Rewrite $\left(2^t\right)\left(2^{3t}\right)$ as a single exponential expression.
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$\left(2^t\right)\left(2^{3t}\right) = 2^{4t}$. Product rule: $2^t \cdot 2^{3t} = 2^{t+3t} = 2^{4t}$.
$\left(2^t\right)\left(2^{3t}\right) = 2^{4t}$. Product rule: $2^t \cdot 2^{3t} = 2^{t+3t} = 2^{4t}$.
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Rewrite $\left(1.5^2\right)^t$ as a single exponential expression with base $1.5$.
Rewrite $\left(1.5^2\right)^t$ as a single exponential expression with base $1.5$.
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$\left(1.5^2\right)^t = 1.5^{2t}$. Power of a power rule: $(1.5^2)^t = 1.5^{2t}$.
$\left(1.5^2\right)^t = 1.5^{2t}$. Power of a power rule: $(1.5^2)^t = 1.5^{2t}$.
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Find and correct the error: $\left(1.10^t\right)^{12} = 1.10^{t+12}$.
Find and correct the error: $\left(1.10^t\right)^{12} = 1.10^{t+12}$.
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Correct: $\left(1.10^t\right)^{12} = 1.10^{12t}$. Power of a power rule multiplies exponents, not adds them.
Correct: $\left(1.10^t\right)^{12} = 1.10^{12t}$. Power of a power rule multiplies exponents, not adds them.
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Identify the equivalent weekly factor if the annual factor is $0.80$.
Identify the equivalent weekly factor if the annual factor is $0.80$.
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$0.80^{\frac{1}{52}}$. Weekly factor is the 52nd root of the annual factor.
$0.80^{\frac{1}{52}}$. Weekly factor is the 52nd root of the annual factor.
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Identify the equivalent monthly factor if the annual factor is $1.06$.
Identify the equivalent monthly factor if the annual factor is $1.06$.
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$1.06^{\frac{1}{12}}$. Monthly factor is the 12th root of the annual factor.
$1.06^{\frac{1}{12}}$. Monthly factor is the 12th root of the annual factor.
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Identify the equivalent quarterly factor if the annual factor is $1.12$.
Identify the equivalent quarterly factor if the annual factor is $1.12$.
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$1.12^{\frac{1}{4}}$. Quarterly factor is the 4th root of the annual factor.
$1.12^{\frac{1}{4}}$. Quarterly factor is the 4th root of the annual factor.
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Rewrite $\left(1.01\right)^{30t}$ as an equivalent factor raised to $t$.
Rewrite $\left(1.01\right)^{30t}$ as an equivalent factor raised to $t$.
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$\left(1.01\right)^{30t} = \left(1.01^{30}\right)^t$. Power of a power rule: $(1.01^{30})^t = 1.01^{30t}$.
$\left(1.01\right)^{30t} = \left(1.01^{30}\right)^t$. Power of a power rule: $(1.01^{30})^t = 1.01^{30t}$.
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Rewrite $\left(0.99\right)^{365t}$ as an equivalent annual factor raised to $t$.
Rewrite $\left(0.99\right)^{365t}$ as an equivalent annual factor raised to $t$.
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$\left(0.99\right)^{365t} = \left(0.99^{365}\right)^t$. Power of a power rule: $(0.99^{365})^t = 0.99^{365t}$.
$\left(0.99\right)^{365t} = \left(0.99^{365}\right)^t$. Power of a power rule: $(0.99^{365})^t = 0.99^{365t}$.
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Rewrite $\left(1.02\right)^{12t}$ as an equivalent annual factor raised to $t$.
Rewrite $\left(1.02\right)^{12t}$ as an equivalent annual factor raised to $t$.
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$\left(1.02\right)^{12t} = \left(1.02^{12}\right)^t$. Power of a power rule: $(1.02^{12})^t = 1.02^{12t}$.
$\left(1.02\right)^{12t} = \left(1.02^{12}\right)^t$. Power of a power rule: $(1.02^{12})^t = 1.02^{12t}$.
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Rewrite $1.44^t$ to show a monthly factor raised to $12t$.
Rewrite $1.44^t$ to show a monthly factor raised to $12t$.
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$1.44^t = \left(1.44^{\frac{1}{12}}\right)^{12t}$. Rewrites annual factor as monthly factor raised to 12 times the power.
$1.44^t = \left(1.44^{\frac{1}{12}}\right)^{12t}$. Rewrites annual factor as monthly factor raised to 12 times the power.
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Rewrite $1.21^t$ to show a quarterly factor raised to $4t$.
Rewrite $1.21^t$ to show a quarterly factor raised to $4t$.
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$1.21^t = \left(1.21^{\frac{1}{4}}\right)^{4t}$. Rewrites annual factor as quarterly factor raised to 4 times the power.
$1.21^t = \left(1.21^{\frac{1}{4}}\right)^{4t}$. Rewrites annual factor as quarterly factor raised to 4 times the power.
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Rewrite $4^{3t}$ as a power with exponent $t$.
Rewrite $4^{3t}$ as a power with exponent $t$.
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$4^{3t} = 64^t$. Since $4^3 = 64$, using power of a power rule gives $(4^3)^t = 64^t$.
$4^{3t} = 64^t$. Since $4^3 = 64$, using power of a power rule gives $(4^3)^t = 64^t$.
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Rewrite $1.08^t$ to show a monthly factor raised to $12t$.
Rewrite $1.08^t$ to show a monthly factor raised to $12t$.
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$1.08^t = \left(1.08^{\frac{1}{12}}\right)^{12t}$. Rewrites annual factor as monthly factor raised to 12 times the power.
$1.08^t = \left(1.08^{\frac{1}{12}}\right)^{12t}$. Rewrites annual factor as monthly factor raised to 12 times the power.
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What is the monthly factor $m$ for a $15%$ annual rate, written exactly using exponents?
What is the monthly factor $m$ for a $15%$ annual rate, written exactly using exponents?
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$m = 1.15^{\frac{1}{12}}$. 15% annually means factor 1.15, so monthly is $1.15^{1/12}$.
$m = 1.15^{\frac{1}{12}}$. 15% annually means factor 1.15, so monthly is $1.15^{1/12}$.
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Identify the annual factor $a$ if the annual interest rate is $R$ (as a decimal).
Identify the annual factor $a$ if the annual interest rate is $R$ (as a decimal).
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$a = 1 + R$. Growth factor is 1 plus the interest rate.
$a = 1 + R$. Growth factor is 1 plus the interest rate.
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Identify the monthly interest rate $r$ if the monthly factor is $m$.
Identify the monthly interest rate $r$ if the monthly factor is $m$.
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$r = m - 1$. Interest rate is the growth factor minus 1.
$r = m - 1$. Interest rate is the growth factor minus 1.
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Rewrite $\left(\frac{16}{25}\right)^{\frac{t}{2}}$ as a base raised to $t$.
Rewrite $\left(\frac{16}{25}\right)^{\frac{t}{2}}$ as a base raised to $t$.
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$\left(\frac{16}{25}\right)^{\frac{t}{2}} = \left(\frac{4}{5}\right)^t$. Since $16/25 = (4/5)^2$, we get $((4/5)^2)^{t/2} = (4/5)^t$.
$\left(\frac{16}{25}\right)^{\frac{t}{2}} = \left(\frac{4}{5}\right)^t$. Since $16/25 = (4/5)^2$, we get $((4/5)^2)^{t/2} = (4/5)^t$.
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Rewrite $\left(\frac{9}{4}\right)^{\frac{t}{2}}$ as a base raised to $t$.
Rewrite $\left(\frac{9}{4}\right)^{\frac{t}{2}}$ as a base raised to $t$.
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$\left(\frac{9}{4}\right)^{\frac{t}{2}} = \left(\frac{3}{2}\right)^t$. Since $9/4 = (3/2)^2$, we get $((3/2)^2)^{t/2} = (3/2)^t$.
$\left(\frac{9}{4}\right)^{\frac{t}{2}} = \left(\frac{3}{2}\right)^t$. Since $9/4 = (3/2)^2$, we get $((3/2)^2)^{t/2} = (3/2)^t$.
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Rewrite $\left(\frac{1}{4}\right)^{\frac{t}{2}}$ as a base raised to $t$.
Rewrite $\left(\frac{1}{4}\right)^{\frac{t}{2}}$ as a base raised to $t$.
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$\left(\frac{1}{4}\right)^{\frac{t}{2}} = \left(\frac{1}{2}\right)^t$. Since $1/4 = (1/2)^2$, we get $((1/2)^2)^{t/2} = (1/2)^t$.
$\left(\frac{1}{4}\right)^{\frac{t}{2}} = \left(\frac{1}{2}\right)^t$. Since $1/4 = (1/2)^2$, we get $((1/2)^2)^{t/2} = (1/2)^t$.
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Rewrite $\left(\frac{1}{3}\right)^{2t}$ as a base raised to $t$.
Rewrite $\left(\frac{1}{3}\right)^{2t}$ as a base raised to $t$.
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$\left(\frac{1}{3}\right)^{2t} = \left(\frac{1}{9}\right)^t$. Since $(1/3)^2 = 1/9$, using power rule gives $(1/9)^t$.
$\left(\frac{1}{3}\right)^{2t} = \left(\frac{1}{9}\right)^t$. Since $(1/3)^2 = 1/9$, using power rule gives $(1/9)^t$.
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Rewrite $32^{\frac{t}{5}}$ as a power with exponent $t$.
Rewrite $32^{\frac{t}{5}}$ as a power with exponent $t$.
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$32^{\frac{t}{5}} = 2^t$. Since $32 = 2^5$, we get $(2^5)^{t/5} = 2^t$.
$32^{\frac{t}{5}} = 2^t$. Since $32 = 2^5$, we get $(2^5)^{t/5} = 2^t$.
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