2. find the value of each variable that makes the equation true.
a. $3^4 \cdot 3^2 = 3^a$
b. $\frac{5^4}{5^3} = 5^b$
c. $4^c = 1$
d. $2^6 \cdot d^6 = 14^6$
e. $6^f = \frac{1}{6}$

Question

Accepted Answer

### Part a
# Explanation:
## Step1: Use exponent product rule
When multiplying exponents with the same base, we add the exponents: $a^m \cdot a^n = a^{m + n}$. So for $3^4 \cdot 3^2$, we have $3^{4+2}$.
$3^4 \cdot 3^2 = 3^{4 + 2}=3^6$
## Step2: Equate exponents
Since $3^4 \cdot 3^2 = 3^a$ and $3^4 \cdot 3^2 = 3^6$, then $3^6=3^a$. For exponential equations with the same base, the exponents must be equal, so $a = 6$.
# Answer:
$a = 6$

### Part b
# Explanation:
## Step1: Use exponent quotient rule
When dividing exponents with the same base, we subtract the exponents: $\frac{a^m}{a^n}=a^{m - n}$. So for $\frac{5^4}{5^3}$, we have $5^{4-3}$.
$\frac{5^4}{5^3}=5^{4 - 3}=5^1$
## Step2: Equate exponents
Since $\frac{5^4}{5^3}=5^b$ and $\frac{5^4}{5^3}=5^1$, then $5^1 = 5^b$. So $b=1$.
# Answer:
$b = 1$

### Part c
# Explanation:
## Step1: Recall the zero - exponent rule
Any non - zero number raised to the power of 0 is 1, i.e., $a^0 = 1$ for $a
eq0$. We have $4^c=1$, and since $4
eq0$, we can use the zero - exponent rule.
## Step2: Solve for $c$
If $4^c = 1$ and $a^0=1$ (with $a = 4$), then $c = 0$ because $4^0=1$.
# Answer:
$c = 0$

### Part d
# Explanation:
## Step1: Use exponent product rule (reverse)
We know that $a^n\cdot b^n=(a\cdot b)^n$. So $2^6\cdot d^6=(2\times d)^6$.
## Step2: Equate the bases
Since $2^6\cdot d^6 = 14^6$ and $2^6\cdot d^6=(2d)^6$, we have $(2d)^6=14^6$. For exponential equations with the same exponent, the bases must be equal (since the exponent is 6, which is even, but we can also consider the positive solution as we are dealing with real numbers in the context of basic exponents), so $2d=14$.
## Step3: Solve for $d$
Divide both sides of $2d = 14$ by 2: $d=\frac{14}{2}=7$.
# Answer:
$d = 7$

### Part e
# Explanation:
## Step1: Rewrite $\frac{1}{6}$ as a power of 6
We know that $\frac{1}{a}=a^{-1}$, so $\frac{1}{6}=6^{-1}$.
## Step2: Equate exponents
Since $6^f=\frac{1}{6}$ and $\frac{1}{6}=6^{-1}$, we have $6^f = 6^{-1}$. For exponential equations with the same base, the exponents must be equal, so $f=-1$.
# Answer:
$f=-1$

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QUESTION IMAGE

Question

Explanation:

Part a

Step1: Use exponent product rule

Step2: Equate exponents

Step1: Use exponent quotient rule

Step2: Equate exponents

Step1: Recall the zero - exponent rule

Step2: Solve for \(c\)

Answer:

Part b