Sure, let's break down each exponential calculation step by step:
1. [tex]\( 3^3 \)[/tex]:
The base is 3 and the exponent is 3, which means we multiply the base by itself three times:
[tex]\[
3 \times 3 \times 3 = 9 \times 3 = 27
\][/tex]
So, [tex]\( 3^3 = 27 \)[/tex].
2. [tex]\( 4^6 \)[/tex]:
The base is 4 and the exponent is 6, which means we multiply the base by itself six times:
[tex]\[
4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096
\][/tex]
So, [tex]\( 4^6 = 4096 \)[/tex].
3. [tex]\( 2^{10} \)[/tex]:
The base is 2 and the exponent is 10, which means we multiply the base by itself ten times:
[tex]\[
2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024
\][/tex]
So, [tex]\( 2^{10} = 1024 \)[/tex].
4. [tex]\( 10^4 \)[/tex]:
The base is 10 and the exponent is 4, which means we multiply the base by itself four times:
[tex]\[
10 \times 10 \times 10 \times 10 = 10000
\][/tex]
So, [tex]\( 10^4 = 10000 \)[/tex].
5. [tex]\( 10^3 \)[/tex]:
The base is 10 and the exponent is 3, which means we multiply the base by itself three times:
[tex]\[
10 \times 10 \times 10 = 1000
\][/tex]
So, [tex]\( 10^3 = 1000 \)[/tex].
6. [tex]\( 10^{-6} \)[/tex]:
A negative exponent indicates a reciprocal, so [tex]\( 10^{-6} \)[/tex] is the same as:
[tex]\[
\frac{1}{10^6} = \frac{1}{1000000} = 0.000001
\][/tex]
Alternatively, this can be written in scientific notation as:
[tex]\[
1 \times 10^{-6} \text{ or simply } 1e-06
\][/tex]
So, [tex]\( 10^{-6} = 1e-06 \)[/tex].
7. [tex]\( 10^{14} \)[/tex]:
The base is 10 and the exponent is 14, which means we multiply the base by itself fourteen times:
[tex]\[
10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 100000000000000
\][/tex]
So, [tex]\( 10^{14} = 100000000000000 \)[/tex].
Here are all the results summarized:
1. [tex]\( 3^3 = 27 \)[/tex]
2. [tex]\( 4^6 = 4096 \)[/tex]
3. [tex]\( 2^{10} = 1024 \)[/tex]
4. [tex]\( 10^4 = 10000 \)[/tex]
5. [tex]\( 10^3 = 1000 \)[/tex]
6. [tex]\( 10^{-6} = 1e-06 \)[/tex]
7. [tex]\( 10^{14} = 100000000000000 \)[/tex]
