To determine which term is a perfect square of the root [tex]\( 3x^4 \)[/tex], we need to find the term that, when squared, yields a perfect square expression. Here are the steps to solve the problem:
1. Identify the root expression: [tex]\( 3x^4 \)[/tex].
- Since we need to address the root term, we need to identify an expression that, when squared, equals [tex]\( 3x^4 \)[/tex].
2. Determine the expression:
- We need to find an expression [tex]\( a \)[/tex] such that [tex]\( a^2 = 3x^4 \)[/tex].
- Take the square root of both sides:
[tex]\[
a = \sqrt{3x^4} = \sqrt{3} \cdot \sqrt{x^4} = \sqrt{3} \cdot x^2
\][/tex]
- So, the root term is [tex]\( \sqrt{3} \cdot x^2 \)[/tex].
3. Check the given terms:
- We need to check whether each of the given terms [tex]\( T \)[/tex] is the square of [tex]\( \sqrt{3} \cdot x^2 \)[/tex].
4. Compare each term:
- Option 1: [tex]\( T = 6x^8 \)[/tex]
- Check if [tex]\( 6x^8 \)[/tex] is a perfect square.
- Write it as [tex]\( ( \sqrt{6} \cdot x^4)^2 \)[/tex].
- Since [tex]\( ( \sqrt{3} \cdot x^2 )^2 = 3 x^4 \)[/tex], [tex]\( 6 x^8 \)[/tex] cannot be the perfect square of [tex]\( (\sqrt{3} \cdot x^2) \)[/tex].
- Option 2: [tex]\( T = 6x^{16} \)[/tex]
- Check if [tex]\( 6x^{16} \)[/tex] is a perfect square.
- Write it as [tex]\( (\sqrt{6} \cdot x^8)^2 \)[/tex].
- Again, since [tex]\( (\sqrt{3} \cdot x^2)^2 = 3x^4 \)[/tex], [tex]\( 6x^{16} \)[/tex] is not the perfect square of [tex]\( (\sqrt{3} \cdot x^2) \)[/tex].
- Option 3: [tex]\( T = 9x^8 \)[/tex]
- Check if [tex]\( 9x^8 \)[/tex] is a perfect square.
- Write it as [tex]\( (3 \cdot x^4)^2 \)[/tex].
- Since [tex]\( (3 x^4)^2 = 9 x^8 \)[/tex], [tex]\( 9x^8 \)[/tex] can be written as a perfect square of [tex]\( (3x^4) \)[/tex]. Therefore, [tex]\( 9x^8 \)[/tex] can be a potential perfect square term for [tex]\( 3x^4 \)[/tex].
- Option 4: [tex]\( T = 9x^{16} \)[/tex]
- Check if [tex]\( 9x^{16} \)[/tex] is a perfect square.
- Write it as [tex]\( (3 \cdot x^8)^2 \)[/tex].
- Since [tex]\( (\sqrt{9 x^{16}})^2 = 3^2 \cdot (x^8)^2 = 9 x^{16} \)[/tex], [tex]\( 9x^{16} \)[/tex] is exact the square of [tex]\( 3 x^8 \)[/tex].
Given these calculations, we can see that the term [tex]\( 9 x^{16} \)[/tex] satisfies the condition of being a perfect square.
Hence, the answer is:
[tex]\[
9 x^{16}
\][/tex]
