Once you ask, “3 to the power of what equals 1/9?” the answer is -2. In mathematical form, this is written as:
3⁻² = 1/9
This result comes from the rule of negative exponents, which shows that a number raised to a negative power equals the reciprocal of that number raised to the matching positive power But it adds up..
Introduction
The question “3 to the power of what equals 1/9?” is a common exponent problem. At first, it may feel confusing because most people are more familiar with positive powers such as:
- 3¹ = 3
- 3² = 9
- 3³ = 27
Still, the answer to this question involves a negative exponent. Negative exponents do not mean the result is negative. Instead, they show that the value is a fraction, specifically the reciprocal of a positive power.
So, if:
3ˣ = 1/9
then:
x = -2
That means 3 to the power of -2 equals 1/9.
Understanding the Problem
The expression can be written as:
3ˣ = 1/9
Here, 3 is the base, x is the exponent, and 1/9 is the result. The goal is to find the exponent that makes the equation true Which is the point..
To solve it, notice that:
9 = 3²
So:
1/9 = 1/3²
Using the negative exponent rule:
1/3² = 3⁻²
Therefore:
3ˣ = 3⁻²
Since the bases are the same, the exponents must also be the same:
x = -2
Step-by-Step Solution
To find what power of 3 equals 1/9, follow these steps:
Step 1: Write the equation
Start with the original question:
3ˣ = 1/9
Step 2: Express 9 as a power of 3
Since:
9 = 3 × 3
then:
9 = 3²
Step 3: Rewrite 1/9 using powers of 3
If:
9 = 3²
then:
1/9 = 1/3²
Step 4: Use the negative exponent rule
The negative exponent rule states:
a⁻ⁿ = 1/aⁿ
So:
1/3² = 3⁻²
Step 5: Compare both sides
Now the equation becomes:
3ˣ = 3⁻²
Because the base is the same on both sides, the exponents must be equal:
x = -2
So the complete answer is:
3⁻² = 1/9
The Negative Exponent Rule
The key idea behind this problem is the negative exponent rule. This rule says that when a number is raised to a negative exponent, the result is the reciprocal of the same number raised to the positive exponent.
In general:
a⁻ⁿ = 1/aⁿ
For example:
- 2⁻¹ = 1/2
- 2⁻² = 1/4
- 5⁻¹ = 1/5
- 5⁻³ = 1/125
- 3⁻² = 1/9
A negative exponent does not make the answer negative. Instead, it tells you to “flip” the number into a fraction.
For example:
3⁻²
does not mean:
-3²
Instead, it means:
1/3²
And since:
3² = 9
then:
3⁻² = 1/9
Why 3 to the Power of 2 Is Not the Answer
A common mistake is to think that because 3² = 9, the answer might be 2. Even so, the question asks for 1/9, not 9 Most people skip this — try not to..
Compare the two:
3² = 9
but:
3⁻² = 1/9
The difference actually matters more than it seems. Even so, a positive exponent gives a whole number or larger value when the base is greater than 1. A negative exponent gives a fraction less than 1 Not complicated — just consistent..
So:
- 3² means multiply 3 by itself: 3 × 3 = 9
- 3⁻² means take the reciprocal of 3²: 1/9
Another Way to Solve It Using Logarithms
You can also solve the problem using logarithms.
Start with:
3ˣ = 1/9
Take the logarithm base 3 of both sides:
log₃(3ˣ) = log₃(1/9)
Using the logarithm rule:
logₐ(aˣ) = x
the left side becomes:
x = log₃(1/9)
Now rewrite 1/9 as a power of 3:
1/9 = 3⁻²
So:
x = log₃(3⁻²)
Again, using the logarithm rule:
x = -2
This confirms the same answer:
3⁻² = 1/9
Examples of Similar Problems
Understanding this one example helps with many similar exponent questions. Here are a few related examples Simple, but easy to overlook..
