$$ \left. \begin{array} { l } { ( \sqrt[ 5 ] { 4 x ^ { 3 } } ) ^ { 15 } } \ { [ ? ] x } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { ( \sqrt[ 5 ] { 4 x ^ { 3 } } ) ^ { 15 } } \ { [ ? ] x } \end{array} ight. $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\left(\sqrt[5]{4x^{3}}\right)^{15}$$
- step1: Simplify the root:
  $$\left(4x^{3}\right)^{3}$$
- step2: Use the properties of exponents:
  $$4^{3}\left(x^{3}\right)^{3}$$
- step3: Evaluate the power:
  $$64x^{9}$$
To simplify the expression $$ \left( \sqrt[5]{4x^{3}} \right)^{15} $$, we can follow these steps:

1. **Apply the power of a power rule**: When raising a power to another power, we multiply the exponents. Thus, we can rewrite the expression as:
   $$
   \left( \sqrt[5]{4} \cdot \sqrt[5]{x^{3}} \right)^{15} = \left( 4^{1/5} \cdot x^{3/5} \right)^{15}
   $$

2. **Distribute the exponent**: Now we apply the exponent of 15 to both parts:
   $$
   4^{1/5 \cdot 15} \cdot x^{3/5 \cdot 15} = 4^{3} \cdot x^{9}
   $$

3. **Calculate $$ 4^{3} $$**: This equals $$ 64 $$.

Putting it all together, we have:
$$
\left( \sqrt[5]{4x^{3}} \right)^{15} = 64x^{9}
$$

Thus, the simplified expression is $$ 64x^{9} $$.
The simplified expression is $$ 64x^{9} $$.

\( \left. \begin{array} { l } { ( \sqrt[ 5 ] { 4 x ^ { 3 } } ) ^ { 15 } } \\ { [ ? ] x } \end{array} \right. \)

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