Use the like bases property to solve the equation

Ask by Guzman Warner. in the United States

Mar 13,2025

Question

Use the like bases property to solve the equation $$ \left(\frac{1}{3}\right)^{n}=243 $$

UpStudy AI · Accepted Answer

To solve the equation $$ \left(\frac{1}{3}\right)^{n}=243 $$ using the like bases property, we first need to express both sides of the equation with the same base.

1. **Identify the base on the left side**: The left side is $$ \left(\frac{1}{3}\right)^{n} $$. We can rewrite $$ \frac{1}{3} $$ as $$ 3^{-1} $$. Therefore, we have:
   $$
   \left(\frac{1}{3}\right)^{n} = (3^{-1})^{n} = 3^{-n}
   $$

2. **Rewrite the right side**: Next, we need to express $$ 243 $$ as a power of $$ 3 $$. We know that:
   $$
   243 = 3^5
   $$

3. **Set the exponents equal**: Now we can rewrite the equation as:
   $$
   3^{-n} = 3^5
   $$
   Since the bases are the same, we can set the exponents equal to each other:
   $$
   -n = 5
   $$

4. **Solve for $$ n $$**: To find $$ n $$, we multiply both sides by -1:
   $$
   n = -5
   $$

Thus, the solution to the equation $$ \left(\frac{1}{3}\right)^{n}=243 $$ is:
$$
\boxed{-5}
$$
$$ n = -5 $$

Use the like bases property to solve the equation

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