To simplify the radical expression [tex]\( \sqrt[4]{48 x^3} \)[/tex] in rational exponent form, follow these steps:
1. Understand the Expression:
The expression [tex]\( \sqrt[4]{48 x^3} \)[/tex] can be rewritten using rational exponents. The fourth root of any number or expression is the same as raising that number or expression to the power of [tex]\( \frac{1}{4} \)[/tex]. Therefore,
[tex]\[
\sqrt[4]{48 x^3} = (48 x^3)^{\frac{1}{4}}
\][/tex]
2. Factorize the Constant:
Break 48 into its prime factors. Since [tex]\( 48 = 2^4 \times 3 \)[/tex], the expression can be rewritten as:
[tex]\[
(48 x^3)^{\frac{1}{4}} = (2^4 \times 3 \times x^3)^{\frac{1}{4}}
\][/tex]
3. Distribute the Exponent:
Apply the exponent [tex]\( \frac{1}{4} \)[/tex] to each factor inside the parentheses individually:
[tex]\[
(2^4 \times 3 \times x^3)^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} \times 3^{\frac{1}{4}} \times (x^3)^{\frac{1}{4}}
\][/tex]
4. Simplify Each Term:
- For [tex]\( (2^4)^{\frac{1}{4}} \)[/tex], apply the power of a power rule: [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex]:
[tex]\[
(2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2
\][/tex]
- The term [tex]\( 3^{\frac{1}{4}} \)[/tex] remains as is since it is already in simplified form.
- For [tex]\( (x^3)^{\frac{1}{4}} \)[/tex], again apply the power of a power rule:
[tex]\[
(x^3)^{\frac{1}{4}} = x^{3 \cdot \frac{1}{4}} = x^{\frac{3}{4}}
\][/tex]
5. Combine the Simplified Terms:
Put all the simplified terms together:
[tex]\[
2 \times 3^{\frac{1}{4}} \times x^{\frac{3}{4}}
\][/tex]
6. Write the Final Simplified Expression:
The simplified rational exponent form of [tex]\( \sqrt[4]{48 x^3} \)[/tex] is:
[tex]\[
2 \times 3^{\frac{1}{4}} \times x^{\frac{3}{4}}
\][/tex]
Answer:
[tex]\[
2 \times 3^{\frac{1}{4}} \times x^{\frac{3}{4}}
\][/tex]
