Question
closed system under multiplication?(1 point)
Responses
(x−−√)(6x⋅−3)
open paren square root of x close paren times open paren 6 x times negative 3 close paren
3(12x)
3 times open paren 1 half x close paren
(x2+1)−(x+7)
open paren x squared plus 1 close paren minus open paren x plus 7 close paren
16x3⋅5
Responses
(x−−√)(6x⋅−3)
open paren square root of x close paren times open paren 6 x times negative 3 close paren
3(12x)
3 times open paren 1 half x close paren
(x2+1)−(x+7)
open paren x squared plus 1 close paren minus open paren x plus 7 close paren
16x3⋅5
Answers
GPT-4o mini
To determine if the given expressions form a closed system under multiplication, we need to check whether the product of any two elements in the set remains in the set and that the set is closed under that operation.
Given the expressions you've provided:
1. \( (\sqrt{x})(6x \cdot -3) \)
2. \( 3 \left( \frac{1}{2} x \right) \)
3. \( (x^2 + 1) - (x + 7) \)
4. \( 16x^3 \cdot 5 \)
**Analysis:**
1. **First Expression: \( (\sqrt{x})(6x \cdot -3) \)**:
- This results in \( -18x^{3/2} \). This expression is still an expression involving \( x \), as long as \( x \geq 0 \) because of the square root.
2. **Second Expression: \( 3 \left( \frac{1}{2} x \right) \)**:
- This simplifies to \( \frac{3}{2} x \). It remains in the form of an expression in \( x \).
3. **Third Expression: \( (x^2 + 1) - (x + 7) \)**:
- This simplifies to \( x^2 - x - 6 \), which is a polynomial in \( x \).
4. **Fourth Expression: \( 16x^3 \cdot 5 \)**:
- This results in \( 80x^3 \), which again is a polynomial in \( x \).
**Conclusion:**
All resulting expressions upon multiplication are still polynomials or algebraic expressions involving \( x \). Therefore, the system is closed under multiplication if we restrict \( x \) to certain values (e.g., \( x \geq 0\) to handle the square roots properly). Since all the expressions remain within the set of algebraic expressions with respect to multiplication, we can say:
**Yes, it forms a closed system under multiplication.**
Given the expressions you've provided:
1. \( (\sqrt{x})(6x \cdot -3) \)
2. \( 3 \left( \frac{1}{2} x \right) \)
3. \( (x^2 + 1) - (x + 7) \)
4. \( 16x^3 \cdot 5 \)
**Analysis:**
1. **First Expression: \( (\sqrt{x})(6x \cdot -3) \)**:
- This results in \( -18x^{3/2} \). This expression is still an expression involving \( x \), as long as \( x \geq 0 \) because of the square root.
2. **Second Expression: \( 3 \left( \frac{1}{2} x \right) \)**:
- This simplifies to \( \frac{3}{2} x \). It remains in the form of an expression in \( x \).
3. **Third Expression: \( (x^2 + 1) - (x + 7) \)**:
- This simplifies to \( x^2 - x - 6 \), which is a polynomial in \( x \).
4. **Fourth Expression: \( 16x^3 \cdot 5 \)**:
- This results in \( 80x^3 \), which again is a polynomial in \( x \).
**Conclusion:**
All resulting expressions upon multiplication are still polynomials or algebraic expressions involving \( x \). Therefore, the system is closed under multiplication if we restrict \( x \) to certain values (e.g., \( x \geq 0\) to handle the square roots properly). Since all the expressions remain within the set of algebraic expressions with respect to multiplication, we can say:
**Yes, it forms a closed system under multiplication.**