1. **State the problem:** We have a number $x$ with a partial prime factorization shown in a tree diagram. We want to find the minimum value of $x$ if $x$ is a perfect cube. 2. **Analyze the prime factorization:** From the tree: - $x = a \times b$ - $a = 2 \times 3$ - $b = 2 \times c$ - $c = 2 \times d$ So, the full prime factorization of $x$ is: $$x = 2 \times 3 \times 2 \times 2 \times d = 2^3 \times 3 \times d$$ 3. **Condition for a perfect cube:** For $x$ to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of 3. 4. **Express $d$ in terms of prime factors:** Since $d$ is a prime factor or product of primes, to minimize $x$, assume $d = 3^k$ for some integer $k$ (since 3 is already a prime factor). 5. **Combine exponents:** The prime factorization is: $$x = 2^3 \times 3^{1+k}$$ 6. **Make exponents multiples of 3:** - The exponent of 2 is already 3, which is a multiple of 3. - The exponent of 3 is $1 + k$, which must be a multiple of 3. 7. **Find minimum $k$:** $$1 + k \equiv 0 \pmod{3} \Rightarrow k = 2$$ 8. **Calculate minimum $x$:** $$x = 2^3 \times 3^{1+2} = 2^3 \times 3^3 = 8 \times 27 = 216$$ **Final answer:** The minimum value of $x$ if $x$ is a perfect cube is **216**.