Answer :
To simplify the expression [tex]\(\left(\sqrt[4]{a^5 b^3}\right)^2\)[/tex], we will follow these steps:
1. Simplify Inside the Radical:
[tex]\[ \sqrt[4]{a^5 b^3} \][/tex]
This can be rewritten using fractional exponents:
[tex]\[ (a^5 b^3)^{1/4} \][/tex]
2. Distribute the Exponent:
Using the property [tex]\((xy)^n = x^n y^n\)[/tex], we distribute the [tex]\(\frac{1}{4}\)[/tex] exponent:
[tex]\[ \left(a^5\right)^{1/4} \left(b^3\right)^{1/4} \][/tex]
Simplify the individual terms:
[tex]\[ a^{5/4} \cdot b^{3/4} \][/tex]
3. Raise the Entire Expression to the Power of 2:
Now take the expression [tex]\(a^{5/4} \cdot b^{3/4}\)[/tex] and raise it to the power of 2:
[tex]\[ \left(a^{5/4} b^{3/4}\right)^2 \][/tex]
4. Distribute the Power:
Using the property [tex]\((x^m)^n = x^{mn}\)[/tex], we distribute the exponent of 2:
[tex]\[ (a^{5/4})^2 \cdot (b^{3/4})^2 \][/tex]
This results in:
[tex]\[ a^{(5/4) \cdot 2} \cdot b^{(3/4) \cdot 2} \][/tex]
5. Simplify the Exponents:
Perform the multiplication in the exponents:
[tex]\[ a^{5/2} \cdot b^{3/2} \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \boxed{a^{5/2} b^{3/2}} \][/tex]
From the provided options, the correct one is:
c. [tex]\(a^{\frac{5}{2}} b^{\frac{3}{2}}\)[/tex]
1. Simplify Inside the Radical:
[tex]\[ \sqrt[4]{a^5 b^3} \][/tex]
This can be rewritten using fractional exponents:
[tex]\[ (a^5 b^3)^{1/4} \][/tex]
2. Distribute the Exponent:
Using the property [tex]\((xy)^n = x^n y^n\)[/tex], we distribute the [tex]\(\frac{1}{4}\)[/tex] exponent:
[tex]\[ \left(a^5\right)^{1/4} \left(b^3\right)^{1/4} \][/tex]
Simplify the individual terms:
[tex]\[ a^{5/4} \cdot b^{3/4} \][/tex]
3. Raise the Entire Expression to the Power of 2:
Now take the expression [tex]\(a^{5/4} \cdot b^{3/4}\)[/tex] and raise it to the power of 2:
[tex]\[ \left(a^{5/4} b^{3/4}\right)^2 \][/tex]
4. Distribute the Power:
Using the property [tex]\((x^m)^n = x^{mn}\)[/tex], we distribute the exponent of 2:
[tex]\[ (a^{5/4})^2 \cdot (b^{3/4})^2 \][/tex]
This results in:
[tex]\[ a^{(5/4) \cdot 2} \cdot b^{(3/4) \cdot 2} \][/tex]
5. Simplify the Exponents:
Perform the multiplication in the exponents:
[tex]\[ a^{5/2} \cdot b^{3/2} \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \boxed{a^{5/2} b^{3/2}} \][/tex]
From the provided options, the correct one is:
c. [tex]\(a^{\frac{5}{2}} b^{\frac{3}{2}}\)[/tex]
