Answer :
Sure! Let's simplify each expression step by step:
1. Expression 1: [tex]\( x \cdot y^6 \)[/tex]
This expression is already in its simplest form. There's nothing to simplify here because there are no similar bases or exponents to combine.
2. Expression 2: [tex]\( x^9 \cdot y^9 \)[/tex]
Like the previous expression, this one is also already simplified. Both variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex], have the same exponent, but since they are different bases, there's no further simplification to be done.
3. Expression 3: [tex]\( x^8 \cdot y^9 \cdot \sqrt[3]{x^2 \cdot y^3} \)[/tex]
Let's simplify this one step by step:
- The cube root of [tex]\( x^2 \cdot y^3 \)[/tex] is [tex]\((x^2 \cdot y^3)^{1/3}\)[/tex].
- This simplifies to [tex]\( x^{2/3} \cdot y^{1} \)[/tex].
- Now, combine these with the original powers:
- For x: [tex]\( x^8 \cdot x^{2/3} = x^{8 + 2/3} \)[/tex].
- For y: [tex]\( y^9 \cdot y^{1} = y^{9 + 1} \)[/tex].
Therefore, the expression simplifies to [tex]\( x^{8 + 2/3} \cdot y^{9 + 1/3} \)[/tex].
4. Expression 4: [tex]\( 4 \cdot \sqrt[4]{x^2 \cdot y^2} \)[/tex]
Let's simplify this one step by step:
- The fourth root of [tex]\( x^2 \cdot y^2 \)[/tex] is [tex]\((x^2 \cdot y^2)^{1/4}\)[/tex].
- This simplifies to [tex]\( x^{2/4} \cdot y^{2/4} = x^{1/2} \cdot y^{1/2} \)[/tex].
- Therefore, the expression becomes [tex]\( 4 \cdot x^{1/2} \cdot y^{1/2} \)[/tex].
Hence, the simplified forms of the expressions are:
1. [tex]\( x \cdot y^6 \)[/tex]
2. [tex]\( x^9 \cdot y^9 \)[/tex]
3. [tex]\( x^{8 + 2/3} \cdot y^{9 + 1/3} \)[/tex]
4. [tex]\( 4 \cdot x^{0.5} \cdot y^{0.5} \)[/tex]
1. Expression 1: [tex]\( x \cdot y^6 \)[/tex]
This expression is already in its simplest form. There's nothing to simplify here because there are no similar bases or exponents to combine.
2. Expression 2: [tex]\( x^9 \cdot y^9 \)[/tex]
Like the previous expression, this one is also already simplified. Both variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex], have the same exponent, but since they are different bases, there's no further simplification to be done.
3. Expression 3: [tex]\( x^8 \cdot y^9 \cdot \sqrt[3]{x^2 \cdot y^3} \)[/tex]
Let's simplify this one step by step:
- The cube root of [tex]\( x^2 \cdot y^3 \)[/tex] is [tex]\((x^2 \cdot y^3)^{1/3}\)[/tex].
- This simplifies to [tex]\( x^{2/3} \cdot y^{1} \)[/tex].
- Now, combine these with the original powers:
- For x: [tex]\( x^8 \cdot x^{2/3} = x^{8 + 2/3} \)[/tex].
- For y: [tex]\( y^9 \cdot y^{1} = y^{9 + 1} \)[/tex].
Therefore, the expression simplifies to [tex]\( x^{8 + 2/3} \cdot y^{9 + 1/3} \)[/tex].
4. Expression 4: [tex]\( 4 \cdot \sqrt[4]{x^2 \cdot y^2} \)[/tex]
Let's simplify this one step by step:
- The fourth root of [tex]\( x^2 \cdot y^2 \)[/tex] is [tex]\((x^2 \cdot y^2)^{1/4}\)[/tex].
- This simplifies to [tex]\( x^{2/4} \cdot y^{2/4} = x^{1/2} \cdot y^{1/2} \)[/tex].
- Therefore, the expression becomes [tex]\( 4 \cdot x^{1/2} \cdot y^{1/2} \)[/tex].
Hence, the simplified forms of the expressions are:
1. [tex]\( x \cdot y^6 \)[/tex]
2. [tex]\( x^9 \cdot y^9 \)[/tex]
3. [tex]\( x^{8 + 2/3} \cdot y^{9 + 1/3} \)[/tex]
4. [tex]\( 4 \cdot x^{0.5} \cdot y^{0.5} \)[/tex]
