Rewrite Exponential Expressions Using Exponents - Algebra 2
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Rewrite $\left(\left(1+r\right)^{12}\right)^t$ as a single exponent on $1+r$.
Rewrite $\left(\left(1+r\right)^{12}\right)^t$ as a single exponent on $1+r$.
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$\left(1+r\right)^{12t}$. Power of a power rule: $ (a^m)^n = a^{mn} $.
$\left(1+r\right)^{12t}$. Power of a power rule: $ (a^m)^n = a^{mn} $.
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Identify the base after rewriting $b^t$ as $\left(b^{\frac{1}{n}}\right)^{nt}$.
Identify the base after rewriting $b^t$ as $\left(b^{\frac{1}{n}}\right)^{nt}$.
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$b^{\frac{1}{n}}$. The new base after transformation.
$b^{\frac{1}{n}}$. The new base after transformation.
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Rewrite $b^{2t}$ to show $6t$ in the exponent: $b^{2t} = (?)^{6t}$.
Rewrite $b^{2t}$ to show $6t$ in the exponent: $b^{2t} = (?)^{6t}$.
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$\left(b^{\frac{1}{3}}\right)^{6t}$. Rewrite using fractional exponent of $\frac{1}{3}$.
$\left(b^{\frac{1}{3}}\right)^{6t}$. Rewrite using fractional exponent of $\frac{1}{3}$.
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Identify the exponent property used to justify $b^t=\left(b^{\frac{1}{n}}\right)^{nt}$.
Identify the exponent property used to justify $b^t=\left(b^{\frac{1}{n}}\right)^{nt}$.
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$(a^m)^n=a^{mn}$. This property allows rewriting for multiple periods.
$(a^m)^n=a^{mn}$. This property allows rewriting for multiple periods.
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State the exponent rule that rewrites $a^{mn}$ using a power raised to a power.
State the exponent rule that rewrites $a^{mn}$ using a power raised to a power.
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$a^{mn}=(a^m)^n$. Power of a power rule: multiply the exponents.
$a^{mn}=(a^m)^n$. Power of a power rule: multiply the exponents.
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State the exponent rule that rewrites $(a^m)^n$ as a single power of $a$.
State the exponent rule that rewrites $(a^m)^n$ as a single power of $a$.
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$(a^m)^n=a^{mn}$. Power of a power rule: multiply the exponents.
$(a^m)^n=a^{mn}$. Power of a power rule: multiply the exponents.
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State the rule for multiplying same-base powers: $a^m\cdot a^n$.
State the rule for multiplying same-base powers: $a^m\cdot a^n$.
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$a^m\cdot a^n=a^{m+n}$. Product of powers rule: add the exponents.
$a^m\cdot a^n=a^{m+n}$. Product of powers rule: add the exponents.
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State the rule for dividing same-base powers: $\frac{a^m}{a^n}$ for $a\ne 0$.
State the rule for dividing same-base powers: $\frac{a^m}{a^n}$ for $a\ne 0$.
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$\frac{a^m}{a^n}=a^{m-n}$. Quotient of powers rule: subtract the exponents.
$\frac{a^m}{a^n}=a^{m-n}$. Quotient of powers rule: subtract the exponents.
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State the meaning of a zero exponent for $a\ne 0$.
State the meaning of a zero exponent for $a\ne 0$.
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$a^0=1$. Any nonzero number to the zero power equals 1.
$a^0=1$. Any nonzero number to the zero power equals 1.
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State the meaning of a negative exponent for $a\ne 0$.
State the meaning of a negative exponent for $a\ne 0$.
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$a^{-n}=\frac{1}{a^n}$. Negative exponent means reciprocal of positive power.
$a^{-n}=\frac{1}{a^n}$. Negative exponent means reciprocal of positive power.
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State the meaning of a fractional exponent $a^{\frac{1}{n}}$ for $a\ge 0$.
State the meaning of a fractional exponent $a^{\frac{1}{n}}$ for $a\ge 0$.
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$a^{\frac{1}{n}}=\sqrt[n]{a}$. Fractional exponent means nth root.
$a^{\frac{1}{n}}=\sqrt[n]{a}$. Fractional exponent means nth root.
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State the meaning of $a^{\frac{m}{n}}$ for $a\ge 0$ and integers $m,n>0$.
State the meaning of $a^{\frac{m}{n}}$ for $a\ge 0$ and integers $m,n>0$.
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$a^{\frac{m}{n}}=\sqrt[n]{a^m}$. Fractional exponent: raise to m, then take nth root.
$a^{\frac{m}{n}}=\sqrt[n]{a^m}$. Fractional exponent: raise to m, then take nth root.
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What is the standard form for an exponential function with initial value $a$ and factor $b$?
What is the standard form for an exponential function with initial value $a$ and factor $b$?
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$f(t)=ab^t$. Standard exponential form with base $b$ and coefficient $a$.
$f(t)=ab^t$. Standard exponential form with base $b$ and coefficient $a$.
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What base $b$ corresponds to exponential growth in $ab^t$?
What base $b$ corresponds to exponential growth in $ab^t$?
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$b>1$. Base greater than 1 causes exponential growth.
$b>1$. Base greater than 1 causes exponential growth.
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What base $b$ corresponds to exponential decay in $ab^t$?
What base $b$ corresponds to exponential decay in $ab^t$?
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$0<b<1$. Base between 0 and 1 causes exponential decay.
$0<b<1$. Base between 0 and 1 causes exponential decay.
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State the transformation that converts $b^t$ into a form with $n$ periods per unit: $b^t=(,?,)^{nt}$.
State the transformation that converts $b^t$ into a form with $n$ periods per unit: $b^t=(,?,)^{nt}$.
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$b^t=\left(b^{\frac{1}{n}}\right)^{nt}$. Power of a power rule applied to create $n$ periods.
$b^t=\left(b^{\frac{1}{n}}\right)^{nt}$. Power of a power rule applied to create $n$ periods.
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Identify the per-period growth factor if the annual factor is $b$ and there are $n$ compounding periods per year.
Identify the per-period growth factor if the annual factor is $b$ and there are $n$ compounding periods per year.
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$b^{\frac{1}{n}}$. nth root of annual factor gives per-period factor.
$b^{\frac{1}{n}}$. nth root of annual factor gives per-period factor.
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Identify the per-period rate $r$ if the per-period factor is $k$.
Identify the per-period rate $r$ if the per-period factor is $k$.
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$r=k-1$. Rate is factor minus 1.
$r=k-1$. Rate is factor minus 1.
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Identify the per-period factor $k$ if the per-period rate is $r$.
Identify the per-period factor $k$ if the per-period rate is $r$.
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$k=1+r$. Factor is 1 plus the rate.
$k=1+r$. Factor is 1 plus the rate.
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What is the monthly factor equivalent to an annual factor of $1.15$ (symbolic form only)?
What is the monthly factor equivalent to an annual factor of $1.15$ (symbolic form only)?
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$1.15^{\frac{1}{12}}$. 12th root of annual factor gives monthly factor.
$1.15^{\frac{1}{12}}$. 12th root of annual factor gives monthly factor.
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Rewrite $1.15^t$ to show $12$ equal growth periods per year using exponent properties.
Rewrite $1.15^t$ to show $12$ equal growth periods per year using exponent properties.
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$1.15^t=\left(1.15^{\frac{1}{12}}\right)^{12t}$. Power of a power rule creates monthly compounding.
$1.15^t=\left(1.15^{\frac{1}{12}}\right)^{12t}$. Power of a power rule creates monthly compounding.
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Rewrite $b^{3t}$ as a power with exponent $t$ (single-step rewrite).
Rewrite $b^{3t}$ as a power with exponent $t$ (single-step rewrite).
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$b^{3t}=(b^3)^t$. Power of a power rule: $(a^m)^n = a^{mn}$.
$b^{3t}=(b^3)^t$. Power of a power rule: $(a^m)^n = a^{mn}$.
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Rewrite $(b^3)^t$ as a single exponential expression in the form $b^{,?,}$.
Rewrite $(b^3)^t$ as a single exponential expression in the form $b^{,?,}$.
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$b^{3t}$. Power of a power rule: $(a^m)^n = a^{mn}$.
$b^{3t}$. Power of a power rule: $(a^m)^n = a^{mn}$.
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Rewrite $b^{\frac{t}{5}}$ in the form $(,?,)^{t}$ using exponent properties.
Rewrite $b^{\frac{t}{5}}$ in the form $(,?,)^{t}$ using exponent properties.
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$\left(b^{\frac{1}{5}}\right)^t$. Fractional exponent becomes root in the base.
$\left(b^{\frac{1}{5}}\right)^t$. Fractional exponent becomes root in the base.
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Rewrite $(b^{\frac{1}{5}})^t$ as a single power of $b$.
Rewrite $(b^{\frac{1}{5}})^t$ as a single power of $b$.
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$b^{\frac{t}{5}}$. Power of a power rule: $(a^m)^n = a^{mn}$.
$b^{\frac{t}{5}}$. Power of a power rule: $(a^m)^n = a^{mn}$.
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Rewrite $b^{t+7}$ as a product involving $b^t$ using exponent properties.
Rewrite $b^{t+7}$ as a product involving $b^t$ using exponent properties.
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$b^{t+7}=b^t\cdot b^7$. Product rule: $a^m \cdot a^n = a^{m+n}$.
$b^{t+7}=b^t\cdot b^7$. Product rule: $a^m \cdot a^n = a^{m+n}$.
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Rewrite $b^{t-4}$ as a quotient involving $b^t$ using exponent properties.
Rewrite $b^{t-4}$ as a quotient involving $b^t$ using exponent properties.
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$b^{t-4}=\frac{b^t}{b^4}$. Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$.
$b^{t-4}=\frac{b^t}{b^4}$. Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$.
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Rewrite $\frac{b^{t}}{b^{2t}}$ as a single power of $b$.
Rewrite $\frac{b^{t}}{b^{2t}}$ as a single power of $b$.
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$b^{-t}$. Quotient rule: $\frac{b^t}{b^{2t}} = b^{t-2t} = b^{-t}$.
$b^{-t}$. Quotient rule: $\frac{b^t}{b^{2t}} = b^{t-2t} = b^{-t}$.
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Rewrite $b^{-t}$ without negative exponents.
Rewrite $b^{-t}$ without negative exponents.
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$\frac{1}{b^t}$. Negative exponent becomes reciprocal.
$\frac{1}{b^t}$. Negative exponent becomes reciprocal.
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Find the equivalent expression for $\left(1+r\right)^{12t}$ written as $\left(,?,\right)^t$.
Find the equivalent expression for $\left(1+r\right)^{12t}$ written as $\left(,?,\right)^t$.
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$\left(1+r\right)^{12t}=\left(\left(1+r\right)^{12}\right)^t$. Power of a power rule applied in reverse.
$\left(1+r\right)^{12t}=\left(\left(1+r\right)^{12}\right)^t$. Power of a power rule applied in reverse.
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