Question

Order the simplification steps of the expression below using the properties of rational exponents.

√567z yll
(567zy¹¹)
3. 2². 1². (71.2².²)
(81-7). z.
3z²y² (7zy³)
3z²y √7zy²³
(81)
.
(7) (+), y(+1)
(34) 712(2+1), (²+1)
3¹. 7. 2². 2. v.
.
.

Answer 1

The correct order for simplifying the expression
`\(\sqrt[4]{567x^9 \cdot y^(11)}\)` using the properties of rational exponents is:
`1. \((567x^9 \cdot y^(11))^(1/4)\). 2. \(3x^2 \cdot y^2 \cdot (7^(1/4) \cdot x^(1/4) \cdot y^(3/4))\). 3. \((81 \cdot 7)^(1/4) \cdot x^(9/4) \cdot y^(11/4)\). 4. \(3x^2 \cdot y^2 \cdot (7x \cdot y^3)^(1/4)\). 5. \(3x^2 \cdot y^2 \cdot \sqrt[4]{7x \cdot y^3}\). 6. \((81)^(1/4) \cdot (7)^(1/4) \cdot x^(2 + 1/4) \cdot y^(2 + 3/4)\). 7. \((3^4)^(1/4) \cdot 7^(1/4) \cdot x^(2 + 1/4) \cdot y^(2 + 3/4)\). 8. \(3^(1-4) \cdot 7^(1/4) \cdot x^2 \cdot x^(1/4) \cdot y^2 \cdot y^(3/4)\)`

First, we rewrite
`\(\sqrt[4]{567x^9 \cdot y^(11)}\)` as
`\((567x^9 \cdot y^(11))^(1/4)\)`. Then, we apply the rules: simplify the numbers inside the parentheses, distribute the exponent to each term, and combine like terms. This results in expressions with smaller exponents. The correct steps ensure that we handle each part of the expression properly, leading to the simplified form.

1.
`\((567x^9 \cdot y^(11))^(1/4)\)`

2.
`\(3x^2 \cdot y^2 \cdot (7^(1/4) \cdot x^(1/4) \cdot y^(3/4))\)`

3.
`\((81 \cdot 7)^(1/4) \cdot x^(9/4) \cdot y^(11/4)\)`

4.
`\(3x^2 \cdot y^2 \cdot (7x \cdot y^3)^(1/4)\)`

5.
`\(3x^2 \cdot y^2 \cdot \sqrt[4]{7x \cdot y^3}\)`

6.
`\((81)^(1/4) \cdot (7)^(1/4) \cdot x^(2 + 1/4) \cdot y^(2 + 3/4)\)`

7.
`\((3^4)^(1/4) \cdot 7^(1/4) \cdot x^(2 + 1/4) \cdot y^(2 + 3/4)\)`

8.
`\(3^(1-4) \cdot 7^(1/4) \cdot x^2 \cdot x^(1/4) \cdot y^2 \cdot y^(3/4)\)`

These steps correctly simplify the expression using the properties of rational exponents.