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Sagot :
To determine which expressions properly show the prime factorization of the number 90, we need to check each expression individually.
1. Expression: [tex]\(2 \times 3 \times 15\)[/tex]
- Let's simplify: [tex]\(2 \times 3 \times 15\)[/tex]
- Calculation: [tex]\(2 \times 3 = 6\)[/tex], then [tex]\(6 \times 15 = 90\)[/tex]
- This product equals 90. Therefore, expression 1 is correct.
2. Expression: [tex]\(2^2 \times 3 \times 5\)[/tex]
- Let's simplify: [tex]\(2^2 \times 3 \times 5\)[/tex]
- First, calculate [tex]\(2^2 = 4\)[/tex], then [tex]\(4 \times 3 = 12\)[/tex], and finally [tex]\(12 \times 5 = 60\)[/tex]
- This product equals 60, not 90. Therefore, expression 2 is incorrect.
3. Expression: [tex]\(2 \times 3^2 \times 5\)[/tex]
- Let's simplify: [tex]\(2 \times 3^2 \times 5\)[/tex]
- First, calculate [tex]\(3^2 = 9\)[/tex], then [tex]\(2 \times 9 = 18\)[/tex], and finally [tex]\(18 \times 5 = 90\)[/tex]
- This product equals 90. Therefore, expression 3 is correct.
4. Expression: [tex]\(2 \times 5 \times 9\)[/tex]
- Let's simplify: [tex]\(2 \times 5 \times 9\)[/tex]
- First, calculate [tex]\(2 \times 5 = 10\)[/tex], then [tex]\(10 \times 9 = 90\)[/tex]
- This product equals 90. Therefore, expression 4 is correct.
5. Expression: [tex]\(2 \times 3 \times 3 \times 5\)[/tex]
- Let's simplify: [tex]\(2 \times 3 \times 3 \times 5\)[/tex]
- First, calculate [tex]\(2 \times 3 = 6\)[/tex], then [tex]\(6 \times 3 = 18\)[/tex], and finally [tex]\(18 \times 5 = 90\)[/tex]
- This product equals 90. Therefore, expression 5 is correct.
By evaluating each expression, we find that the prime factorizations showing the number 90 correctly are:
[tex]\[ \boxed{2 \times 3 \times 15} \][/tex]
[tex]\[ \boxed{2 \times 3^2 \times 5} \][/tex]
[tex]\[ \boxed{2 \times 5 \times 9} \][/tex]
[tex]\[ \boxed{2 \times 3 \times 3 \times 5} \][/tex]
Thus, the correct expressions from the provided list that show the prime factorization of 90 are: [tex]\(2 \times 3 \times 15\)[/tex], [tex]\(2 \times 3^2 \times 5\)[/tex], [tex]\(2 \times 5 \times 9\)[/tex], and [tex]\(2 \times 3 \times 3 \times 5\)[/tex].
1. Expression: [tex]\(2 \times 3 \times 15\)[/tex]
- Let's simplify: [tex]\(2 \times 3 \times 15\)[/tex]
- Calculation: [tex]\(2 \times 3 = 6\)[/tex], then [tex]\(6 \times 15 = 90\)[/tex]
- This product equals 90. Therefore, expression 1 is correct.
2. Expression: [tex]\(2^2 \times 3 \times 5\)[/tex]
- Let's simplify: [tex]\(2^2 \times 3 \times 5\)[/tex]
- First, calculate [tex]\(2^2 = 4\)[/tex], then [tex]\(4 \times 3 = 12\)[/tex], and finally [tex]\(12 \times 5 = 60\)[/tex]
- This product equals 60, not 90. Therefore, expression 2 is incorrect.
3. Expression: [tex]\(2 \times 3^2 \times 5\)[/tex]
- Let's simplify: [tex]\(2 \times 3^2 \times 5\)[/tex]
- First, calculate [tex]\(3^2 = 9\)[/tex], then [tex]\(2 \times 9 = 18\)[/tex], and finally [tex]\(18 \times 5 = 90\)[/tex]
- This product equals 90. Therefore, expression 3 is correct.
4. Expression: [tex]\(2 \times 5 \times 9\)[/tex]
- Let's simplify: [tex]\(2 \times 5 \times 9\)[/tex]
- First, calculate [tex]\(2 \times 5 = 10\)[/tex], then [tex]\(10 \times 9 = 90\)[/tex]
- This product equals 90. Therefore, expression 4 is correct.
5. Expression: [tex]\(2 \times 3 \times 3 \times 5\)[/tex]
- Let's simplify: [tex]\(2 \times 3 \times 3 \times 5\)[/tex]
- First, calculate [tex]\(2 \times 3 = 6\)[/tex], then [tex]\(6 \times 3 = 18\)[/tex], and finally [tex]\(18 \times 5 = 90\)[/tex]
- This product equals 90. Therefore, expression 5 is correct.
By evaluating each expression, we find that the prime factorizations showing the number 90 correctly are:
[tex]\[ \boxed{2 \times 3 \times 15} \][/tex]
[tex]\[ \boxed{2 \times 3^2 \times 5} \][/tex]
[tex]\[ \boxed{2 \times 5 \times 9} \][/tex]
[tex]\[ \boxed{2 \times 3 \times 3 \times 5} \][/tex]
Thus, the correct expressions from the provided list that show the prime factorization of 90 are: [tex]\(2 \times 3 \times 15\)[/tex], [tex]\(2 \times 3^2 \times 5\)[/tex], [tex]\(2 \times 5 \times 9\)[/tex], and [tex]\(2 \times 3 \times 3 \times 5\)[/tex].
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