To simplify the given mathematical expressions step-by-step, let's work through each one carefully.
### Expression 1: [tex]\(\left(x^2 y^4\right)\left(x y^7\right)\)[/tex]
Step 1: Distribute the terms within the parentheses.
- For the [tex]\(x\)[/tex] terms: [tex]\(x^2 \cdot x\)[/tex]
- For the [tex]\(y\)[/tex] terms: [tex]\(y^4 \cdot y^7\)[/tex]
Step 2: Apply the laws of exponents. When multiplying powers with the same base, you add the exponents.
- [tex]\(x^2 \cdot x = x^{2+1} = x^3\)[/tex]
- [tex]\(y^4 \cdot y^7 = y^{4+7} = y^{11}\)[/tex]
So, the simplified form of [tex]\(\left(x^2 y^4\right)\left(x y^7\right)\)[/tex] is:
[tex]\[ x^3 y^{11} \][/tex]
### Expression 2: [tex]\(\frac{z}{y^{-3}}\)[/tex]
Step 1: Simplify the negative exponent. Recall that [tex]\(y^{-n} = \frac{1}{y^n}\)[/tex].
[tex]\[ \frac{z}{y^{-3}} = z \cdot y^3 \][/tex]
So, the simplified form of [tex]\(\frac{z}{y^{-3}}\)[/tex] is:
[tex]\[ zy^3 \][/tex]
### Expression 3: [tex]\(\frac{z}{y^3}\)[/tex]
This expression is already in its simplest form and cannot be simplified further.
[tex]\[ \frac{z}{y^3} \][/tex]
### Expression 4: [tex]\(x^3 y^{11}\)[/tex]
This expression is already in its simplest form as well.
[tex]\[ x^3 y^{11} \][/tex]
### Expression 5: [tex]\(x^{11} y^3\)[/tex]
This expression is also already in its simplest form.
[tex]\[ x^{11} y^3 \][/tex]
In summary, the simplified forms of the given expressions are:
- [tex]\(\left(x^2 y^4\right)\left(x y^7\right) \)[/tex] simplifies to [tex]\(x^3 y^{11}\)[/tex]
- [tex]\(\frac{z}{y^{-3}}\)[/tex] simplifies to [tex]\(zy^3\)[/tex]
- [tex]\(\frac{z}{y^3}\)[/tex] is already simplified as [tex]\(\frac{z}{y^3}\)[/tex]
- [tex]\(x^3 y^{11}\)[/tex] is already simplified as [tex]\(x^3 y^{11}\)[/tex]
- [tex]\(x^{11} y^3\)[/tex] is already simplified as [tex]\(x^{11} y^3\)[/tex]
Therefore, the correct answer to simplify [tex]\(\left(x^2 y^4\right)\left(x y^7\right)\)[/tex] is:
[tex]\[ x^3 y^{11} \][/tex]
