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Answer :
To simplify the expression [tex]\(\sqrt{x^5 y^5} \cdot 4 \sqrt{3 x^9 y^8}\)[/tex], follow these steps:
1. Understand Square Roots and Multiplication:
- The expression involves two square roots and a constant multiplication factor.
2. Combine the Square Roots:
- We have [tex]\(\sqrt{x^5 y^5}\)[/tex] and [tex]\(\sqrt{3 x^9 y^8}\)[/tex].
- Using the property [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex], we can combine them under one square root.
- [tex]\( \sqrt{x^5 y^5} \cdot \sqrt{3 x^9 y^8} = \sqrt{x^5 y^5 \cdot 3 x^9 y^8}\)[/tex].
3. Simplify Inside the Square Root:
- Multiply inside the square root:
- [tex]\(x^5 \cdot x^9 = x^{5+9} = x^{14}\)[/tex].
- [tex]\(y^5 \cdot y^8 = y^{5+8} = y^{13}\)[/tex].
- So, inside the square root, we have [tex]\(3 \cdot x^{14} \cdot y^{13}\)[/tex].
4. Write the Expression as Simplified:
- The expression so far is [tex]\(4 \cdot \sqrt{3 \cdot x^{14} \cdot y^{13}}\)[/tex].
5. Final Simplified Expression:
- There are no further reductions possible for the entire expression in this form because the exponents do not allow taking simpler powers out of the root easily.
- Remember that multiplying outside the root doesn't change this simplified version and hence it remains [tex]\(4\sqrt{3} \sqrt{x^5 y^5} \sqrt{x^9 y^8}\)[/tex].
Thus, the simplified expression is [tex]\(4\sqrt{3} \sqrt{x^5 y^5} \sqrt{x^9 y^8}\)[/tex].
1. Understand Square Roots and Multiplication:
- The expression involves two square roots and a constant multiplication factor.
2. Combine the Square Roots:
- We have [tex]\(\sqrt{x^5 y^5}\)[/tex] and [tex]\(\sqrt{3 x^9 y^8}\)[/tex].
- Using the property [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex], we can combine them under one square root.
- [tex]\( \sqrt{x^5 y^5} \cdot \sqrt{3 x^9 y^8} = \sqrt{x^5 y^5 \cdot 3 x^9 y^8}\)[/tex].
3. Simplify Inside the Square Root:
- Multiply inside the square root:
- [tex]\(x^5 \cdot x^9 = x^{5+9} = x^{14}\)[/tex].
- [tex]\(y^5 \cdot y^8 = y^{5+8} = y^{13}\)[/tex].
- So, inside the square root, we have [tex]\(3 \cdot x^{14} \cdot y^{13}\)[/tex].
4. Write the Expression as Simplified:
- The expression so far is [tex]\(4 \cdot \sqrt{3 \cdot x^{14} \cdot y^{13}}\)[/tex].
5. Final Simplified Expression:
- There are no further reductions possible for the entire expression in this form because the exponents do not allow taking simpler powers out of the root easily.
- Remember that multiplying outside the root doesn't change this simplified version and hence it remains [tex]\(4\sqrt{3} \sqrt{x^5 y^5} \sqrt{x^9 y^8}\)[/tex].
Thus, the simplified expression is [tex]\(4\sqrt{3} \sqrt{x^5 y^5} \sqrt{x^9 y^8}\)[/tex].
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