What exponent would make the two expressions equivalent: (3^3)^x = 3^24?
To solve the equation (3^3)^x = 3^24, we can use the property of exponents that states (a^b)^c = a^(b*c). Applying this property to the left side of the equation, we have 3^(3*x) = 3^24.
In order for two expressions with the same base to be equivalent, their exponents must be equal. Therefore, we set 3*x = 24.
Solving the equation for x, we divide both sides by 3:
3*x/3 = 24/3
x = 8
Hence, the exponent that makes the two expressions equivalent is x = 8.
To find the exponent that would make the two expressions equivalent, you need to equate the bases and set the exponents equal to each other.
We have:
(3^3)^x = 3^24
First, simplify the left side by applying the power of a power rule:
3^(3x) = 3^24
Now, since the bases are equal (which is 3), we can equate the exponents:
3x = 24
To find the value of x, divide both sides of the equation by 3:
x = 24/3
Simplifying further,
x = 8
Therefore, the exponent that would make the two expressions equivalent is x = 8.
To find the exponent that would make the two expressions equivalent, you can start by simplifying each expression individually.
First, simplify (3^3)^x. This can be done by raising the base (3^3) to the power of x. Using the exponent rule for exponentiation, we can multiply the exponents together: (3^3)^x = 3^(3x).
Next, let's simplify 3^24. This expression can be evaluated by raising the base 3 to the power of 24: 3^24 = 282,429,536,481.
Now we have two equivalent expressions: 3^(3x) = 282,429,536,481.
To find the value of x, we need to solve this equation for x. We can do this by taking the logarithm of both sides of the equation. Let's take the logarithm base 3 (log₃) of both sides:
log₃(3^(3x)) = log₃(282,429,536,481).
Using the power rule of logarithms, the exponent can be brought down as a coefficient:
3x · log₃(3) = log₃(282,429,536,481).
Since log₃(3) is equal to 1 (because 3^1 = 3), the equation simplifies to:
3x = log₃(282,429,536,481).
Now, we can divide both sides of the equation by 3 to solve for x:
x = log₃(282,429,536,481) / 3.
Using a calculator or software that can compute logarithms, we can find that log₃(282,429,536,481) is approximately 11. So the simplified value of x is:
x = 11 / 3.
Hence, the exponent that would make the two expressions equivalent is 11/3.