Answer :
To determine which expressions are equivalent to \(3^2 \cdot 3^4\), let's start by simplifying \(3^2 \cdot 3^4\).
Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 3^2 \cdot 3^4 = 3^{2+4} = 3^6 \][/tex]
So, we need to check which of the given expressions simplify to \(3^6\):
1. \(3^6\):
- This is already in the form \(3^6\), so it is equivalent.
2. \(3^8\):
- This is \(3^8\), which is not equivalent to \(3^6\).
3. \(9^6\):
- \(9\) can be written as \(3^2\), so \(9^6 = (3^2)^6 = 3^{2 \cdot 6} = 3^{12}\), which is not equivalent to \(3^6\).
4. \(3^{-4} \cdot 3^{10}\):
- Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 3^{-4} \cdot 3^{10} = 3^{-4+10} = 3^6 \][/tex]
- This is equivalent to \(3^6\).
5. \(3^0 \cdot 3^8\):
- Using the property of exponents:
[tex]\[ 3^0 \cdot 3^8 = 3^{0+8} = 3^8 \][/tex]
- This is not equivalent to \(3^6\).
6. \(3^3 \cdot 3^3\):
- Using the property of exponents:
[tex]\[ 3^3 \cdot 3^3 = 3^{3+3} = 3^6 \][/tex]
- This is equivalent to \(3^6\).
7. \((3^2) \cdot (3^4)\):
- This is the same as the original expression \(3^2 \cdot 3^4\), which simplifies to \(3^6\).
- This is equivalent to \(3^6\).
8. \((3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3)\):
- \((3 \cdot 3)\) can be written as \(3^2\), and \((3 \cdot 3 \cdot 3 \cdot 3)\) can be written as \(3^4\):
[tex]\[ (3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3) = 3^2 \cdot 3^4 = 3^6 \][/tex]
- This is equivalent to \(3^6\).
Therefore, the expressions that are equivalent to \(3^2 \cdot 3^4\) are:
- \(3^6\)
- \(3^{-4} \cdot 3^{10}\)
- \(3^3 \cdot 3^3\)
- \((3 \cdot 3) \cdot(3 \cdot 3 \cdot 3 \cdot 3)\)
Thus, the equivalent expressions are numbers 1, 4, 6, and 8.
Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 3^2 \cdot 3^4 = 3^{2+4} = 3^6 \][/tex]
So, we need to check which of the given expressions simplify to \(3^6\):
1. \(3^6\):
- This is already in the form \(3^6\), so it is equivalent.
2. \(3^8\):
- This is \(3^8\), which is not equivalent to \(3^6\).
3. \(9^6\):
- \(9\) can be written as \(3^2\), so \(9^6 = (3^2)^6 = 3^{2 \cdot 6} = 3^{12}\), which is not equivalent to \(3^6\).
4. \(3^{-4} \cdot 3^{10}\):
- Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 3^{-4} \cdot 3^{10} = 3^{-4+10} = 3^6 \][/tex]
- This is equivalent to \(3^6\).
5. \(3^0 \cdot 3^8\):
- Using the property of exponents:
[tex]\[ 3^0 \cdot 3^8 = 3^{0+8} = 3^8 \][/tex]
- This is not equivalent to \(3^6\).
6. \(3^3 \cdot 3^3\):
- Using the property of exponents:
[tex]\[ 3^3 \cdot 3^3 = 3^{3+3} = 3^6 \][/tex]
- This is equivalent to \(3^6\).
7. \((3^2) \cdot (3^4)\):
- This is the same as the original expression \(3^2 \cdot 3^4\), which simplifies to \(3^6\).
- This is equivalent to \(3^6\).
8. \((3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3)\):
- \((3 \cdot 3)\) can be written as \(3^2\), and \((3 \cdot 3 \cdot 3 \cdot 3)\) can be written as \(3^4\):
[tex]\[ (3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3) = 3^2 \cdot 3^4 = 3^6 \][/tex]
- This is equivalent to \(3^6\).
Therefore, the expressions that are equivalent to \(3^2 \cdot 3^4\) are:
- \(3^6\)
- \(3^{-4} \cdot 3^{10}\)
- \(3^3 \cdot 3^3\)
- \((3 \cdot 3) \cdot(3 \cdot 3 \cdot 3 \cdot 3)\)
Thus, the equivalent expressions are numbers 1, 4, 6, and 8.
