Solve for x: 5^(x+2) = 1/25 and 27.3^x = 1

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the equations.

3.3.1 Solve: $27^{3x} = 1$

Express the right side of the equation as a power of 27.
Any non-zero number raised to the power of 0 is 1.
$27^{3x} = 27^0$
Since the bases are the same, set the exponents equal to each other.
$3x = 0$
Solve for $x$ .
$x = \frac{0}{3}$ $x = 0$
The final answer is $\boxed{\mathbf{0}}$ .

3.3.2 Solve: $5^{x+2} = \frac{1}{25}$
Express the right side of the equation as a power of 5.
$25 = 5^2$
So,
$\frac{1}{25} = \frac{1}{5^2}$
Use the rule $a^{-n} = \frac{1}{a^n}$ to rewrite $\frac{1}{5^2}$ .
$\frac{1}{5^2} = 5^{-2}$
Substitute this back into the original equation.
$5^{x+2} = 5^{-2}$
Since the bases are the same, set the exponents equal to each other.
$x+2 = -2$
Solve for $x$ .
$x = -2 - 2$ $x = -4$
The final answer is $\boxed{\mathbf{-4}}$ .

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