Question

Solve each equation. $$ 3^{-x}=81 $$

Short answer

x = -4

Step-by-step solution

- Rewrite 81 as a power of 3

First, recognize that 81 can be written as a power of 3. We know that 81 = 3^4.

- Set the exponents equal to each other

Next, rewrite the equation 3^{-x} = 81 as 3^{-x} = 3^4. Since the bases are the same, we can set the exponents equal to each other:-x = 4.

- Solve for x

To solve for x, simply divide both sides of the equation -x = 4 by -1:x = -4.

Key concepts

Powers of Numbers

Understanding powers, or exponents, of numbers is crucial. When we say a number is raised to a certain power, like in our exercise, it means the number is multiplied by itself a specific number of times. For instance:

• 3^2 means 3 multiplied by itself 2 times: 3 × 3 = 9.
• 3^3 means 3 multiplied by itself 3 times: 3 × 3 × 3 = 27.
• Similarly, 3^4 means 3 × 3 × 3 × 3 = 81.

In our exercise, we need to recognize that 81 can be written as 3^4.
This helps us simplify and solve the equation. Remembering these powers can save you time when you're working on similar problems.

Solving Equations

Solving equations like the given exercise involves several steps. The main goal is to isolate the variable (in this case, x) to find its value. Let’s break down the process:
First, we simplify the equation by rewriting 81 as a power of 3, which we already know is 3^4. Now, our equation looks like 3^{-x} = 3^4.
Since the bases are the same, we can set the exponents equal. This gives us: -x = 4.
Finally, to solve for x, we need to isolate it. We do this by dividing both sides by -1, leading us to x = -4.
Remember to always check your solution by plugging it back into the original equation to make sure it works. For negative exponents, we’ll be covering that next.

Negative Exponents

Negative exponents might seem tricky, but they follow a simple rule. A negative exponent indicates that you divide 1 by the base raised to the exponent. Here’s how it works:

• For example, 3^{-2} means 1 divided by 3^2, which is 1/9.
• Similarly, 3^{-3} means 1 divided by 3^3, which equals 1/27.

In our exercise, we dealt with 3^{-x}. When we set -x equal to 4,
we find x = -4. This shows how negative exponents help us solve certain equations effectively.
Becoming comfortable with negative exponents means knowing they reverse multiplication into division.
Always remember: the negative sign in the exponent tells you to take the reciprocal of the base raised to the positive exponent.