To solve the expression [tex]\(\left(-2 \sqrt[3]{x^3 y}\right)\left(-5 \sqrt[4]{x y^2}\right)\left(3 \sqrt[12]{x^3 y^2}\right)\)[/tex], let's break it down step-by-step:
1. Define each term in the expression:
- First term: [tex]\(-2 \sqrt[3]{x^3 y}\)[/tex]
- Second term: [tex]\(-5 \sqrt[4]{x y^2}\)[/tex]
- Third term: [tex]\(3 \sqrt[12]{x^3 y^2}\)[/tex]
2. Rewrite each term with fractional exponents:
- [tex]\( \sqrt[3]{x^3 y} \)[/tex] is equivalent to [tex]\((x^3 y)^{\frac{1}{3}}\)[/tex]
- [tex]\( \sqrt[4]{x y^2} \)[/tex] is equivalent to [tex]\((x y^2)^{\frac{1}{4}}\)[/tex]
- [tex]\( \sqrt[12]{x^3 y^2} \)[/tex] is equivalent to [tex]\((x^3 y^2)^{\frac{1}{12}}\)[/tex]
Now the terms are:
- First term: [tex]\(-2 (x^3 y)^{\frac{1}{3}}\)[/tex]
- Second term: [tex]\(-5 (x y^2)^{\frac{1}{4}}\)[/tex]
- Third term: [tex]\( 3 (x^3 y^2)^{\frac{1}{12}} \)[/tex]
3. Combine the terms by multiplication:
[tex]\[ \left(-2 (x^3 y)^{\frac{1}{3}}\right) \left(-5 (x y^2)^{\frac{1}{4}}\right) \left(3 (x^3 y^2)^{\frac{1}{12}}\right) \][/tex]
4. Multiply the constants:
Combine [tex]\(-2\)[/tex], [tex]\(-5\)[/tex], and [tex]\(3\)[/tex]:
[tex]\[ (-2) \times (-5) \times 3 = 30 \][/tex]
5. Combine the exponent terms:
When multiplying terms with the same base, you add the exponents. So you need to combine the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- For [tex]\(x\)[/tex]:
[tex]\[
x^{\frac{3}{3}} \times x^{\frac{1}{4}} \times x^{\frac{3}{12}}
\][/tex]
Rewrite the exponents with a common denominator (which is 12):
[tex]\[
x^1 \times x^{\frac{3}{12}} \times x^{\frac{1}{4}} = x^{\frac{12}{12}} \times x^{\frac{1}{4}} = x^{1 + \frac{3}{12} + \frac{1}{4}} = x^{1 + \frac{3}{12} + \frac{3}{12}} = x^{1 + \frac{6}{12}} = x^{1.5}
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
y^{\frac{1}{3}} \times y^{\frac{2}{4}} \times y^{\frac{2}{12}}
\][/tex]
Rewrite the exponents with a common denominator (which is 12):
[tex]\[
y^{\frac{4}{12}} \times y^{\frac{6}{12}} \times y^{\frac{2}{12}} = y^{\frac{4 + 6 + 2}{12}} = y^{\frac{12}{12}} = y^1
\][/tex]
6. Combine the results:
Now put everything together:
[tex]\[
30 x^{1.5} y
\][/tex]
7. Simplify the final result
It can be rewritten as:
[tex]\[
30 x^{3/2} y
\][/tex]
So, the simplified form of the expression [tex]\(\left(-2 \sqrt[3]{x^3 y}\right)\left(-5 \sqrt[4]{x y^2}\right)\left(3 \sqrt[12]{x^3 y^2}\right)\)[/tex] is [tex]\(30 x^{3/2} y\)[/tex].
