Question

kuta software - infinite precalculus
name
exponential equations not requiring logarithms
date
p
solve each equation.

1. $5^{3n}=125$
2. $2^{2n}=16$
3. $5^{3r}=5^{-2r}$
4. $3^{-2k}=81$
5. $2^{-3x}=2^{x - 1}$
6. $6^{3m}=36$
7. $10^{3x}=10000$
8. $4^{r + 1}=1$
9. $32^{2x}=8$
10. $\left(\frac{1}{6}\

ight)^n = 36$

Explanation:

Step1: Rewrite as same base

$5^{3n}=5^3$

Step2: Equate exponents

$3n=3$

Step3: Solve for n

$n=\frac{3}{3}=1$

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Step1: Rewrite as same base

$2^{5n}=2^4$

Step2: Equate exponents

$5n=4$

Step3: Solve for n

$n=\frac{4}{5}$

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Step1: Equate exponents (same base)

$3r=-2r$

Step2: Isolate r term

$3r+2r=0$

Step3: Solve for r

$5r=0 \implies r=0$

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Step1: Rewrite as same base

$3^{-2k}=3^4$

Step2: Equate exponents

$-2k=4$

Step3: Solve for k

$k=\frac{4}{-2}=-2$

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Step1: Equate exponents (same base)

$-3x=x-1$

Step2: Isolate x terms

$-3x-x=-1$

Step3: Solve for x

$-4x=-1 \implies x=\frac{1}{4}$

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Step1: Rewrite as same base

$6^{3m}=6^2$

Step2: Equate exponents

$3m=2$

Step3: Solve for m

$m=\frac{2}{3}$

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Step1: Rewrite as same base

$10^{3x}=10^4$

Step2: Equate exponents

$3x=4$

Step3: Solve for x

$x=\frac{4}{3}$

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Step1: Rewrite as same base

$4^{r+1}=4^0$

Step2: Equate exponents

$r+1=0$

Step3: Solve for r

$r=-1$

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Step1: Rewrite as base 5

$(2^5)^{2x}=2^3 \implies 2^{10x}=2^3$

Step2: Equate exponents

$10x=3$

Step3: Solve for x

$x=\frac{3}{10}$

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Step1: Rewrite as same base

$(6^{-1})^n=6^2 \implies 6^{-n}=6^2$

Step2: Equate exponents

$-n=2$

Step3: Solve for n

$n=-2$

Answer:

1. $n=1$
2. $n=\frac{4}{5}$
3. $r=0$
4. $k=-2$
5. $x=\frac{1}{4}$
6. $m=\frac{2}{3}$
7. $x=\frac{4}{3}$
8. $r=-1$
9. $x=\frac{3}{10}$
10. $n=-2$