Use the like bases property to solve the equation

Ask by Watson Ford. in the United States

Mar 13,2025

Question

Use the like bases property to solve the equation $$ \left(\frac{1}{2}\right)^{n+1}=2^{5 n-5} $$ $$ n=\square $$

UpStudy AI · Accepted Answer

We start with the equation

$$
\left(\frac{1}{2}\right)^{n+1} = 2^{5n-5}.
$$

Express $$\frac{1}{2}$$ as $$2^{-1}$$:

$$
\left(2^{-1}\right)^{n+1} = 2^{5n-5}.
$$

Simplify the left side using the power of a power property:

$$
2^{- (n+1)} = 2^{5n-5}.
$$

Since the bases are the same, we set the exponents equal:

$$
-(n+1) = 5n - 5.
$$

Distribute the negative sign on the left:

$$
- n - 1 = 5n - 5.
$$

Add $$n$$ to both sides:

$$
-1 = 6n - 5.
$$

Add $$5$$ to both sides:

$$
4 = 6n.
$$

Finally, divide by $$6$$:

$$
n = \frac{4}{6} = \frac{2}{3}.
$$
$$ n = \frac{2}{3} $$

Use the like bases property to solve the equation