ANSWER
10a²b³
EXPLANATION
To simplify this expression, we will use the following properties of exponents:
• Radicals are fractional exponents:
[tex]\sqrt[b]{a}=a^{1/b}[/tex]• Product of powers with the same base:
[tex]a^b\cdot a^c=a^{b+c}[/tex]• Power of a power:
[tex](a^b)^c=a^{b\cdot c}[/tex]• Exponents can be distributed into the product:
[tex](a\cdot b)^c=a^c\cdot b^c[/tex]First, write the radicals as fractional exponents,
[tex]\sqrt[]{a^3b^5}\cdot5\sqrt[]{4ab}=(a^3b^5)^{1/2}\cdot5(4ab)^{1/2}[/tex]Distribute the fractional exponents into each product,
[tex](a^3b^5)^{1/2}\cdot5(4ab)^{1/2}=(a^3)^{1/2}(b^5)^{1/2}\cdot5(4)^{1/2}(a)^{1/2}(b)^{1/2}[/tex]Solve the power of the constants - in this case, the only constant is 4, and also, apply the rule of the power of a power for the first two factors,
[tex](a^3)^{1/2}(b^5)^{1/2}\cdot5(4)^{1/2}(a)^{1/2}(b)^{1/2}=a^{3/2}\cdot b^{5/2}\cdot5\cdot2\cdot a^{1/2}\cdot b^{1/2}[/tex]Solve the product between the constants (5*2) and apply the rule of powers with the same base for a and b,
[tex](a^{3/2}\cdot a^{1/2})\cdot(b^{5/2}\cdot b^{1/2})\cdot(5\cdot2)=(a^{3/2+1/2})\cdot(b^{5/2+1/2})\cdot(10)[/tex]Solve the additions in the exponents and write the constant first,
[tex](a^{3/2+1/2})\cdot(b^{5/2+1/2})\cdot(10)=10\cdot a^2\cdot b^3[/tex]Hence, the simplified expression is 10a²b³ .
