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Sagot :
Sure, let's simplify each expression step by step.
### 1. Simplify [tex]\(\sqrt{20 x^{12} y^6}\)[/tex]:
To simplify the square root of a product involving variables, we can break it down step by step:
[tex]\[ \sqrt{20 x^{12} y^6} \][/tex]
First, we need to write [tex]\(20\)[/tex] as a product of its prime factors:
[tex]\[ 20 = 2^2 \cdot 5 \][/tex]
Thus, we can rewrite the expression as:
[tex]\[ \sqrt{2^2 \cdot 5 \cdot x^{12} \cdot y^6} \][/tex]
Since the square root of a product is the product of the square roots, we have:
[tex]\[ \sqrt{2^2} \cdot \sqrt{5} \cdot \sqrt{x^{12}} \cdot \sqrt{y^6} \][/tex]
We know:
[tex]\[ \sqrt{2^2} = 2 \][/tex]
[tex]\[ \sqrt{5} = \sqrt{5} \][/tex]
[tex]\[ \sqrt{x^{12}} = x^{12/2} = x^6 \][/tex]
[tex]\[ \sqrt{y^6} = y^{6/2} = y^3 \][/tex]
Putting it all together, we get:
[tex]\[ 2 \sqrt{5} \cdot x^6 \cdot y^3 \][/tex]
Therefore, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^6 y^3} \][/tex]
### 2. Simplify [tex]\(10 x^6 |y^3|\)[/tex]:
This expression is already simplified, but let's rewrite it for clarity:
[tex]\[ 10 x^6 |y^3| \][/tex]
Therefore, it remains:
[tex]\[ \boxed{10 x^6 |y^3|} \][/tex]
### 3. Simplify [tex]\(2 x^5 y^2 \sqrt{5 x y}\)[/tex]:
To simplify, let's first address the square root part:
[tex]\[ \sqrt{5 x y} = \sqrt{5} \cdot \sqrt{x} \cdot \sqrt{y} \][/tex]
Putting this back into the expression, we get:
[tex]\[ 2 x^5 y^2 \left( \sqrt{5} \cdot \sqrt{x} \cdot \sqrt{y} \right) \][/tex]
Combine the terms under single powers where possible:
[tex]\[ = 2 \sqrt{5} \cdot x^5 \cdot y^2 \cdot \sqrt{x} \cdot \sqrt{y} \][/tex]
[tex]\[ = 2 \sqrt{5} \cdot x^5 \cdot y^2 \cdot x^{1/2} \cdot y^{1/2} \][/tex]
Combine the exponents:
[tex]\[ = 2 \sqrt{5} \cdot x^{5 + 1/2} \cdot y^{2 + 1/2} \][/tex]
[tex]\[ = 2 \sqrt{5} \cdot x^{11/2} \cdot y^{5/2} \][/tex]
Therefore, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^{11/2} y^{5/2}} \][/tex]
### 4. Simplify [tex]\(2 (x y)^9 \sqrt{5}\)[/tex]:
To simplify this expression, recognize that [tex]\((x y)^9\)[/tex] can be separated:
[tex]\[ 2 (x y)^9 \sqrt{5} = 2 \cdot (x^9 y^9) \cdot \sqrt{5} \][/tex]
Thus, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^9 y^9} \][/tex]
### 5. Simplify [tex]\(2 x^6 |y^3| \sqrt{5}\)[/tex]:
This expression involves the absolute value and the square root of a constant. Let's rewrite it clearly:
[tex]\[ 2 x^6 |y^3| \sqrt{5} \][/tex]
Thus, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^6 |y^3|} \][/tex]
These are the simplified forms of the given expressions.
### 1. Simplify [tex]\(\sqrt{20 x^{12} y^6}\)[/tex]:
To simplify the square root of a product involving variables, we can break it down step by step:
[tex]\[ \sqrt{20 x^{12} y^6} \][/tex]
First, we need to write [tex]\(20\)[/tex] as a product of its prime factors:
[tex]\[ 20 = 2^2 \cdot 5 \][/tex]
Thus, we can rewrite the expression as:
[tex]\[ \sqrt{2^2 \cdot 5 \cdot x^{12} \cdot y^6} \][/tex]
Since the square root of a product is the product of the square roots, we have:
[tex]\[ \sqrt{2^2} \cdot \sqrt{5} \cdot \sqrt{x^{12}} \cdot \sqrt{y^6} \][/tex]
We know:
[tex]\[ \sqrt{2^2} = 2 \][/tex]
[tex]\[ \sqrt{5} = \sqrt{5} \][/tex]
[tex]\[ \sqrt{x^{12}} = x^{12/2} = x^6 \][/tex]
[tex]\[ \sqrt{y^6} = y^{6/2} = y^3 \][/tex]
Putting it all together, we get:
[tex]\[ 2 \sqrt{5} \cdot x^6 \cdot y^3 \][/tex]
Therefore, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^6 y^3} \][/tex]
### 2. Simplify [tex]\(10 x^6 |y^3|\)[/tex]:
This expression is already simplified, but let's rewrite it for clarity:
[tex]\[ 10 x^6 |y^3| \][/tex]
Therefore, it remains:
[tex]\[ \boxed{10 x^6 |y^3|} \][/tex]
### 3. Simplify [tex]\(2 x^5 y^2 \sqrt{5 x y}\)[/tex]:
To simplify, let's first address the square root part:
[tex]\[ \sqrt{5 x y} = \sqrt{5} \cdot \sqrt{x} \cdot \sqrt{y} \][/tex]
Putting this back into the expression, we get:
[tex]\[ 2 x^5 y^2 \left( \sqrt{5} \cdot \sqrt{x} \cdot \sqrt{y} \right) \][/tex]
Combine the terms under single powers where possible:
[tex]\[ = 2 \sqrt{5} \cdot x^5 \cdot y^2 \cdot \sqrt{x} \cdot \sqrt{y} \][/tex]
[tex]\[ = 2 \sqrt{5} \cdot x^5 \cdot y^2 \cdot x^{1/2} \cdot y^{1/2} \][/tex]
Combine the exponents:
[tex]\[ = 2 \sqrt{5} \cdot x^{5 + 1/2} \cdot y^{2 + 1/2} \][/tex]
[tex]\[ = 2 \sqrt{5} \cdot x^{11/2} \cdot y^{5/2} \][/tex]
Therefore, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^{11/2} y^{5/2}} \][/tex]
### 4. Simplify [tex]\(2 (x y)^9 \sqrt{5}\)[/tex]:
To simplify this expression, recognize that [tex]\((x y)^9\)[/tex] can be separated:
[tex]\[ 2 (x y)^9 \sqrt{5} = 2 \cdot (x^9 y^9) \cdot \sqrt{5} \][/tex]
Thus, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^9 y^9} \][/tex]
### 5. Simplify [tex]\(2 x^6 |y^3| \sqrt{5}\)[/tex]:
This expression involves the absolute value and the square root of a constant. Let's rewrite it clearly:
[tex]\[ 2 x^6 |y^3| \sqrt{5} \][/tex]
Thus, the simplified form is:
[tex]\[ \boxed{2 \sqrt{5} x^6 |y^3|} \][/tex]
These are the simplified forms of the given expressions.
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