Expressing 27 as a Power of 3

To express 27 as a power of 3, we need to find the exponent that, when raised to 3, gives us 27. In other words, we are looking for the value of x in the equation 3^x = 27.

Using Prime Factorization
One way to find the exponent is by using prime factorization. We can express 27 as a product of prime numbers and then determine the exponent.

Step 1: Prime Factorization of 27
We start by dividing 27 by its smallest prime factor, which is 3. Since 27 is divisible by 3, we get 27 ÷ 3 = 9.

Step 2: Continue Prime Factorization
Next, we divide 9 by its smallest prime factor, which is also 3. We get 9 ÷ 3 = 3.

Step 3: Final Prime Factorization
Now, we have reached a prime number, which is 3. So, the prime factorization of 27 is 3 × 3 × 3, or simply 3^3.

Therefore, 27 can be expressed as 3 raised to the power of 3, written as 3^3.

Using Logarithms
Another way to find the exponent is by using logarithms. We can take the logarithm of both sides of the equation 3^x = 27.

Step 1: Take the Logarithm
We use the logarithm base 3 on both sides of the equation:
log3(3^x) = log3(27).

Step 2: Apply the Logarithmic Identity
Since the logarithm base 3 of 3^x is simply x, and the logarithm base 3 of 27 is the unknown exponent, we have:
x = log3(27).

Step 3: Evaluate the Logarithm
Using a calculator, we can evaluate the logarithm of 27 base 3. The result is x = 3.

Therefore, 27 can also be expressed as 3 raised to the power of 3, written as 3^3.

Conclusion
In conclusion, 27 can be expressed as 3 raised to the power of 3, which is written as 3^3. This means that 3 multiplied by itself three times equals 27.

The answer is 3^3