To determine whether each of the given equations is true or false, let's assess them one by one.
1. Equation: [tex]\(3^5 = 3 \times 5\)[/tex]
Left-hand side (LHS):
[tex]\(3^5\)[/tex] is the same as multiplying 3 by itself 5 times:
[tex]\[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 \][/tex]
Right-hand side (RHS):
[tex]\(3 \times 5\)[/tex] is simply:
[tex]\[ 3 \times 5 = 15 \][/tex]
Now, we compare the two sides:
The left-hand side is [tex]\(243\)[/tex], and the right-hand side is [tex]\(15\)[/tex]. Clearly, [tex]\(243 \neq 15\)[/tex].
Therefore, the equation [tex]\(3^5 = 3 \times 5\)[/tex] is False.
2. Equation: [tex]\(3^5 = 5^3\)[/tex]
Left-hand side (LHS):
[tex]\(3^5\)[/tex] remains as previously calculated:
[tex]\[ 3^5 = 243 \][/tex]
Right-hand side (RHS):
[tex]\(5^3\)[/tex] is the same as multiplying 5 by itself 3 times:
[tex]\[ 5^3 = 5 \times 5 \times 5 = 125 \][/tex]
Now, we compare the two sides:
The left-hand side is [tex]\(243\)[/tex], and the right-hand side is [tex]\(125\)[/tex]. Clearly, [tex]\(243 \neq 125\)[/tex].
Therefore, the equation [tex]\(3^5 = 5^3\)[/tex] is False.
3. Equation: [tex]\(1^3 = 5^3\)[/tex]
Left-hand side (LHS):
[tex]\(1^3\)[/tex] is the same as multiplying 1 by itself 3 times:
[tex]\[ 1^3 = 1 \][/tex]
Right-hand side (RHS):
[tex]\(5^3\)[/tex] remains as previously calculated:
[tex]\[ 5^3 = 125 \][/tex]
Now, we compare the two sides:
The left-hand side is [tex]\(1\)[/tex], and the right-hand side is [tex]\(125\)[/tex]. Clearly, [tex]\(1 \neq 125\)[/tex].
Therefore, the equation [tex]\(1^3 = 5^3\)[/tex] is False.
Based on the evaluations provided, here are the results for each of the equations:
[tex]\[
\begin{array}{ll}
& \text{True} \quad \text{False} \\
3^5 = 3 \times 5 & \quad False \\
3^5 = 5^3 & \quad False \\
1^3 = 5^3 & \quad False \\
\end{array}
\][/tex]
Hence, all three equations are False.
