n-linear equations. 30. 9^x+1=81 31. 64cdot 4^3x=16 32 2^-2x+5cdot (1)/(4)=2^x+2

## Step 1: Rewrite 81 as a power of 9<br />### We can rewrite 81 as $9^2$. The equation becomes $9^{x+1} = 9^2$.<br /><br />## Step 2: Equate the exponents<br />### Since the bases are equal, the exponents must be equal. Therefore, $x+1 = 2$.<br /><br />## Step 3: Solve for x<br />### Subtract 1 from both sides of the equation: $x + 1 - 1 = 2 - 1$. This simplifies to $x = 1$.<br /><br />## Step 4: Rewrite the equation with a common base<br />### We can rewrite 64 and 16 as powers of 4: $64 = 4^3$ and $16 = 4^2$. The equation becomes $4^3 \cdot 4^{3x} = 4^2$.<br /><br />## Step 5: Simplify the left side using exponent rules<br />### When multiplying exponential terms with the same base, we add the exponents: $4^{3+3x} = 4^2$.<br /><br />## Step 6: Equate the exponents and solve for x<br />### Since the bases are equal, the exponents must be equal: $3 + 3x = 2$. Subtract 3 from both sides: $3x = -1$. Divide by 3: $x = -\frac{1}{3}$.<br /><br />## Step 7: Rewrite the equation with a common base of 2<br />### We can rewrite $\frac{1}{4}$ as $2^{-2}$ and 64 as $2^6$. The equation becomes $2^{-2x+5} \cdot 2^{-2} = 2^{x+2}$.<br /><br />## Step 8: Simplify the left side using exponent rules<br />### When multiplying exponential terms with the same base, we add the exponents: $2^{-2x+5-2} = 2^{x+2}$, which simplifies to $2^{-2x+3} = 2^{x+2}$.<br /><br />## Step 9: Equate the exponents and solve for x<br />### Since the bases are equal, the exponents must be equal: $-2x + 3 = x + 2$. Add $2x$ to both sides: $3 = 3x + 2$. Subtract 2 from both sides: $1 = 3x$. Divide by 3: $x = \frac{1}{3}$.