To determine which of the given numbers is algebraically the largest, we can evaluate them individually. Let's consider the given options step by step:
1. [tex]\( A = -9.7 \times 10^2 \)[/tex]
- Calculating this value:
[tex]\[ -9.7 \times 10^2 = -9.7 \times 100 = -970.0 \][/tex]
2. [tex]\( B = 9.7 \times 10^{-2} \)[/tex]
- Calculating this value:
[tex]\[ 9.7 \times 10^{-2} = 9.7 \times 0.01 = 0.097 \][/tex]
3. [tex]\( C = 9.7 \times 10^3 \)[/tex]
- Calculating this value:
[tex]\[ 9.7 \times 10^3 = 9.7 \times 1000 = 9700.0 \][/tex]
4. [tex]\( D = -9.7 \times 10^{-2} \)[/tex]
- Calculating this value:
[tex]\[ -9.7 \times 10^{-2} = -9.7 \times 0.01 = -0.097 \][/tex]
5. [tex]\( E = 9.7 \times 10^2 \)[/tex]
- Calculating this value:
[tex]\[ 9.7 \times 10^2 = 9.7 \times 100 = 970.0 \][/tex]
Now, let's compare these values to determine the largest one:
- [tex]\( -970.0 \)[/tex]
- [tex]\( 0.097 \)[/tex]
- [tex]\( 9700.0 \)[/tex] (This appears to be significantly larger than the others)
- [tex]\( -0.097 \)[/tex]
- [tex]\( 970.0 \)[/tex]
Clearly, [tex]\( 9700.0 \)[/tex] is algebraically the largest value among all the choices.
Thus, the number that is algebraically the largest is [tex]\( 9.7 \times 10^3 \)[/tex].
Therefore, the correct option is:
[tex]\[ \boxed{C} \][/tex]
The numerical result confirms that the largest value is [tex]\( 9700.0 \)[/tex], and this corresponds to option C.
