Answer :
To find an expression equivalent to [tex]\( 16^{\frac{3}{4} x} \)[/tex], let's compare each option to the given expression. We will simplify each option and compare it to the original expression.
### Given Expression:
[tex]\[ 16^{\frac{3}{4} x} \][/tex]
#### Step 1: Simplify the Given Expression
First, express [tex]\( 16 \)[/tex] as a power of 2:
[tex]\[ 16 = 2^4 \][/tex]
Then, substitute this back into the expression:
[tex]\[ 16^{\frac{3}{4} x} = (2^4)^{\frac{3}{4} x} \][/tex]
Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (2^4)^{\frac{3}{4} x} = 2^{4 \cdot \frac{3}{4} x} = 2^{3x} \][/tex]
So, the given expression simplifies to:
[tex]\[ 2^{3x} \][/tex]
### Now, let's analyze each option:
#### Option 1: [tex]\(\sqrt[4]{16}^{3 x}\)[/tex]
Rewrite [tex]\(\sqrt[4]{16}\)[/tex] as [tex]\(16^{1/4}\)[/tex]:
[tex]\[ \sqrt[4]{16}^{3 x} = (16^{1/4})^{3 x} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/4} = 2^{4 \cdot 1/4} = 2^1 = 2 \][/tex]
Thus:
[tex]\[ (2)^{3 x} = 2^{3 x} \][/tex]
This expression is [tex]\(2^{3x}\)[/tex], which is the same as our simplified given expression. Hence, Option 1 is equivalent.
#### Option 2: [tex]\(\sqrt[4x]{16}\)[/tex]
Rewrite [tex]\(\sqrt[4x]{16}\)[/tex] as [tex]\(16^{1/(4x)}\)[/tex]:
[tex]\[ 16^{1/(4x)} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/(4x)} = 2^{4 \cdot (1/(4x))} = 2^{1/x} \][/tex]
This expression is [tex]\(2^{1/x}\)[/tex], which is not the same as our simplified given expression. Hence, Option 2 is not equivalent.
#### Option 3: [tex]\(\sqrt[3]{16}^{4 x}\)[/tex]
Rewrite [tex]\(\sqrt[3]{16}\)[/tex] as [tex]\(16^{1/3}\)[/tex]:
[tex]\[ \sqrt[3]{16}^{4 x} = (16^{1/3})^{4 x} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/3} = 2^{4 \cdot 1/3} = 2^{4/3} \][/tex]
Thus:
[tex]\[ (2^{4/3})^{4x} = 2^{(4/3) \cdot 4x} = 2^{16x/3} \][/tex]
This expression is [tex]\(2^{16x/3}\)[/tex], which is not the same as our simplified given expression. Hence, Option 3 is not equivalent.
#### Option 4: [tex]\(\sqrt[3x]{16}\)[/tex]
Rewrite [tex]\(\sqrt[3x]{16}\)[/tex] as [tex]\(16^{1/(3x)}\)[/tex]:
[tex]\[ 16^{1/(3x)} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/(3x)} = 2^{4 \cdot (1/(3x))} = 2^{4/(3x)} \][/tex]
This expression is [tex]\(2^{4/(3x)}\)[/tex], which is not the same as our simplified given expression. Hence, Option 4 is not equivalent.
### Conclusion
The expression equivalent to [tex]\( 16^{\frac{3}{4} x} \)[/tex] is:
[tex]\[ \sqrt[4]{16}^{3x} \][/tex]
Thus, the correct answer is Option 1.
### Given Expression:
[tex]\[ 16^{\frac{3}{4} x} \][/tex]
#### Step 1: Simplify the Given Expression
First, express [tex]\( 16 \)[/tex] as a power of 2:
[tex]\[ 16 = 2^4 \][/tex]
Then, substitute this back into the expression:
[tex]\[ 16^{\frac{3}{4} x} = (2^4)^{\frac{3}{4} x} \][/tex]
Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (2^4)^{\frac{3}{4} x} = 2^{4 \cdot \frac{3}{4} x} = 2^{3x} \][/tex]
So, the given expression simplifies to:
[tex]\[ 2^{3x} \][/tex]
### Now, let's analyze each option:
#### Option 1: [tex]\(\sqrt[4]{16}^{3 x}\)[/tex]
Rewrite [tex]\(\sqrt[4]{16}\)[/tex] as [tex]\(16^{1/4}\)[/tex]:
[tex]\[ \sqrt[4]{16}^{3 x} = (16^{1/4})^{3 x} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/4} = 2^{4 \cdot 1/4} = 2^1 = 2 \][/tex]
Thus:
[tex]\[ (2)^{3 x} = 2^{3 x} \][/tex]
This expression is [tex]\(2^{3x}\)[/tex], which is the same as our simplified given expression. Hence, Option 1 is equivalent.
#### Option 2: [tex]\(\sqrt[4x]{16}\)[/tex]
Rewrite [tex]\(\sqrt[4x]{16}\)[/tex] as [tex]\(16^{1/(4x)}\)[/tex]:
[tex]\[ 16^{1/(4x)} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/(4x)} = 2^{4 \cdot (1/(4x))} = 2^{1/x} \][/tex]
This expression is [tex]\(2^{1/x}\)[/tex], which is not the same as our simplified given expression. Hence, Option 2 is not equivalent.
#### Option 3: [tex]\(\sqrt[3]{16}^{4 x}\)[/tex]
Rewrite [tex]\(\sqrt[3]{16}\)[/tex] as [tex]\(16^{1/3}\)[/tex]:
[tex]\[ \sqrt[3]{16}^{4 x} = (16^{1/3})^{4 x} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/3} = 2^{4 \cdot 1/3} = 2^{4/3} \][/tex]
Thus:
[tex]\[ (2^{4/3})^{4x} = 2^{(4/3) \cdot 4x} = 2^{16x/3} \][/tex]
This expression is [tex]\(2^{16x/3}\)[/tex], which is not the same as our simplified given expression. Hence, Option 3 is not equivalent.
#### Option 4: [tex]\(\sqrt[3x]{16}\)[/tex]
Rewrite [tex]\(\sqrt[3x]{16}\)[/tex] as [tex]\(16^{1/(3x)}\)[/tex]:
[tex]\[ 16^{1/(3x)} \][/tex]
Substitute [tex]\(16 = 2^4\)[/tex]:
[tex]\[ (2^4)^{1/(3x)} = 2^{4 \cdot (1/(3x))} = 2^{4/(3x)} \][/tex]
This expression is [tex]\(2^{4/(3x)}\)[/tex], which is not the same as our simplified given expression. Hence, Option 4 is not equivalent.
### Conclusion
The expression equivalent to [tex]\( 16^{\frac{3}{4} x} \)[/tex] is:
[tex]\[ \sqrt[4]{16}^{3x} \][/tex]
Thus, the correct answer is Option 1.
