$$ x^{x^{3}}=27 $$ find $$ 2 x $$

Question

UpStudy AI · Accepted Answer

To solve the equation $$ x^{x^{3}}=27 $$, we can start by rewriting $$ 27 $$ as a power of $$ 3 $$:

$$
27 = 3^3
$$

Thus, we can rewrite the equation as:

$$
x^{x^{3}} = 3^3
$$

Next, we can take the logarithm of both sides. Let's use the natural logarithm (ln):

$$
\ln(x^{x^{3}}) = \ln(3^3)
$$

Using the property of logarithms that states $$ \ln(a^b) = b \ln(a) $$, we can simplify both sides:

$$
x^{3} \ln(x) = 3 \ln(3)
$$

Now, we need to solve for $$ x $$. This equation is not straightforward, so we can try some values for $$ x $$ to see if we can find a solution.

Let's test $$ x = 3 $$:

$$
3^{3} \ln(3) = 3 \ln(3)
$$

Calculating $$ 3^{3} $$:

$$
27 \ln(3) = 3 \ln(3)
$$

This does not hold true. Let's try $$ x = 1 $$:

$$
1^{1^{3}} \ln(1) = 3 \ln(3)
$$

This also does not hold true. Now let's try $$ x = 3^{1/3} $$:

$$
(3^{1/3})^{(3^{1/3})^{3}} = 3
$$

Calculating $$ (3^{1/3})^{(3^{1})} = 3^{1/3} $$:

This does not hold true either.

Let's try $$ x = 3^{1/3} $$:

$$
(3^{1/3})^{(3^{1/3})^{3}} = (3^{1/3})^{3} = 3
$$

This holds true. Therefore, we have found that:

$$
x = 3^{1/3}
$$

Now, we need to find $$ 2x $$:

$$
2x = 2 \cdot 3^{1/3}
$$

Thus, the final answer is:

$$
2x = 2 \cdot 3^{1/3}
$$
$$ 2x = 2 \times 3^{1/3} $$

\( x^{x^{3}}=27 \) find \( 2 x \)