SOLVING AN EXPONENTIAL EQUATIO OMMON BASE $$ \begin{array}{l} =81 \ 3^{x}=3^{4} \ x=4 \end{array} $$ 2 solve $$ 16^{x-2}=4 $$ $$ \begin{array}{l} 16^{x-2}=4 \ \frac{1}{4^{2(x-2)}=4^{\prime}} \ \quad 2(x-2)=1 \end{array} $$ 4 $$ \begin{array}{l} 8^{2 x+3}=1 \ 8^{2 x+3}=8^{0} \rightarrow \frac{2 x+13-5}{2 x=-2} \end{array} $$ $$ 716^{2 x+1}=64^{x+5} $$ $$ 6 \quad 9^{3 x-1} $$ $$ =27 $$ 8 $$ 16^{x+3}=32^{3 x-2} $$ $$ \begin{aligned} 9^{3 x-1}=3^{3} \rightarrow 2(3 x-1) & =3 \ 6 x-1 / 2 & =3 \ 1 & +2(3 x-1) \end{aligned} $$ $$ 16^{x+3}=32^{3 x-2} $$

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UpStudy AI · Accepted Answer

Let's solve the exponential equations step by step.

### Problem 1: Solve $$ 16^{x-2} = 4 $$

1. **Rewrite the bases**: 
   - We know that $$ 16 = 4^2 $$, so we can rewrite the equation as:
   $$
   (4^2)^{x-2} = 4
   $$
   This simplifies to:
   $$
   4^{2(x-2)} = 4^1
   $$

2. **Set the exponents equal**: 
   Since the bases are the same, we can set the exponents equal to each other:
   $$
   2(x-2) = 1
   $$

3. **Solve for $$ x $$**:
   $$
   2x - 4 = 1 \
   2x = 5 \
   x = \frac{5}{2}
   $$

### Problem 2: Solve $$ 8^{2x+3} = 1 $$

1. **Rewrite the equation**: 
   - We know that $$ 1 = 8^0 $$, so we can rewrite the equation as:
   $$
   8^{2x+3} = 8^0
   $$

2. **Set the exponents equal**: 
   $$
   2x + 3 = 0
   $$

3. **Solve for $$ x $$**:
   $$
   2x = -3 \
   x = -\frac{3}{2}
   $$

### Problem 3: Solve $$ 716^{2x+1} = 64^{x+5} $$

1. **Rewrite the bases**: 
   - We know that $$ 716 = 2^{6} $$ and $$ 64 = 2^{6} $$, so we can rewrite the equation as:
   $$
   (2^6)^{2x+1} = (2^6)^{x+5}
   $$
   This simplifies to:
   $$
   2^{6(2x+1)} = 2^{6(x+5)}
   $$

2. **Set the exponents equal**: 
   $$
   6(2x + 1) = 6(x + 5)
   $$

3. **Solve for $$ x $$**:
   $$
   12x + 6 = 6x + 30 \
   12x - 6x = 30 - 6 \
   6x = 24 \
   x = 4
   $$

### Problem 4: Solve $$ 16^{x+3} = 32^{3x-2} $$

1. **Rewrite the bases**: 
   - We know that $$ 16 = 2^4 $$ and $$ 32 = 2^5 $$, so we can rewrite the equation as:
   $$
   (2^4)^{x+3} = (2^5)^{3x-2}
   $$
   This simplifies to:
   $$
   2^{4(x+3)} = 2^{5(3x-2)}
   $$

2. **Set the exponents equal**: 
   $$
   4(x + 3) = 5(3x - 2)
   $$

3. **Solve for $$ x $$**:
   $$
   4x + 12 = 15x - 10 \
   12 + 10 = 15x - 4x \
   22 = 11x \
   x = 2
   $$

### Summary of Solutions:
1. $$ x = \frac{5}{2} $$ for $$ 16^{x-2} = 4 $$
2. $$ x = -\frac{3}{2} $$ for $$ 8^{2x+3} = 1 $$
3. $$ x = 4 $$ for $$ 716^{2x+1} = 64^{x+5} $$
4. $$ x = 2 $$ for $$ 16^{x+3} = 32^{3x-2} $$
Solutions to the exponential equations are: 1. $$ x = \frac{5}{2} $$ 2. $$ x = -\frac{3}{2} $$ 3. $$ x = 4 $$ 4. $$ x = 2 $$

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