Question

2a.6 exponential equations iw
equivalence property if $a^m = a^n$, then $m = n$
same base
$6^{-2m} = 6^2$ $3^{3 - 2m} = 3^4$ $5^{-x} = 5^3$ $3^{-x - 1} = 3^{-x}$ $4^{-3b - 3} = 4^3$

Explanation:

Step1: Set exponents equal (same base)

For $6^{-2m}=6^2$, apply equivalence property:
$-2m = 2$

Step2: Solve for m

Divide both sides by -2:
$m = \frac{2}{-2} = -1$

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Step1: Set exponents equal (same base)

For $3^{3-2m}=3^4$, apply equivalence property:
$3-2m = 4$

Step2: Isolate the term with m

Subtract 3 from both sides:
$-2m = 4-3 = 1$

Step3: Solve for m

Divide both sides by -2:
$m = -\frac{1}{2}$

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Step1: Set exponents equal (same base)

For $5^{-x}=5^3$, apply equivalence property:
$-x = 3$

Step2: Solve for x

Multiply both sides by -1:
$x = -3$

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Step1: Set exponents equal (same base)

For $3^{-x-1}=3^{-x}$, apply equivalence property:
$-x-1 = -x$

Step2: Simplify equation

Add $x$ to both sides:
$-1 = 0$
This is a false statement, so no solution exists.

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Step1: Set exponents equal (same base)

For $4^{-3b-3}=4^3$, apply equivalence property:
$-3b-3 = 3$

Step2: Isolate the term with b

Add 3 to both sides:
$-3b = 3+3 = 6$

Step3: Solve for b

Divide both sides by -3:
$b = \frac{6}{-3} = -2$

Answer:

1. $m = -1$
2. $m = -\frac{1}{2}$
3. $x = -3$
4. No solution
5. $b = -2$