Let's solve these mathematical expressions involving powers of 5 step-by-step.
1. [tex]\( 5^{-3} \cdot 5^{-3} \)[/tex]
- When multiplying powers with the same base, we add the exponents.
- [tex]\( 5^{-3 + (-3)} = 5^{-6} \)[/tex]
2. [tex]\( 5 \cdot 5^3 \)[/tex]
- Again, we add the exponents.
- [tex]\( 5^1 \cdot 5^3 = 5^{1+3} = 5^4 \)[/tex]
3. [tex]\( 5^6 \cdot 5^{-4} \)[/tex]
- Add the exponents.
- [tex]\( 5^{6 + (-4)} = 5^2 \)[/tex]
4. [tex]\( 5^{-4} \cdot 5^4 \cdot 5^0 \)[/tex]
- Add the exponents.
- [tex]\( 5^{-4 + 4 + 0} = 5^0 \)[/tex]
- Any number to the power of 0 is 1.
- So, [tex]\( 5^0 = 1 \)[/tex]
5. [tex]\( 5^7 \cdot 5^3 \)[/tex]
- Add the exponents.
- [tex]\( 5^{7 + 3} = 5^{10} \)[/tex]
6. [tex]\( 5^4 \)[/tex]
- This expression is already in its simplest form.
7. [tex]\( 5^0 \)[/tex]
- Any non-zero number raised to the power of 0 is 1.
- So, [tex]\( 5^0 = 1 \)[/tex]
8. [tex]\( \frac{1}{5^6} \)[/tex]
- This expression is already in its simplest form.
9. [tex]\( 5^2 \)[/tex]
- This expression is already in its simplest form.
10. [tex]\( 5^0 \)[/tex]
- Any non-zero number raised to the power of 0 is 1.
- So, [tex]\( 5^0 = 1 \)[/tex]
Having solved these, we have the following:
1. [tex]\( 5^{-3} \cdot 5^{-3} = 5^{-6} \)[/tex]
2. [tex]\( 5 \cdot 5^3 = 5^4 \)[/tex]
3. [tex]\( 5^6 \cdot 5^{-4} = 5^2 \)[/tex]
4. [tex]\( 5^{-4} \cdot 5^4 \cdot 5^0 = 5^0 = 1 \)[/tex]
5. [tex]\( 5^7 \cdot 5^3 = 5^{10} \)[/tex]
6. [tex]\( 5^4 \)[/tex]
7. [tex]\( 5^0 = 1 \)[/tex]
8. [tex]\( \frac{1}{5^6} \)[/tex]
9. [tex]\( 5^2 \)[/tex]
10. [tex]\( 5^0 = 1 \)[/tex]
So our final answers are:
1. [tex]\( 5^{-3} \cdot 5^{-3} = 5^{-6} \)[/tex]
2. [tex]\( 5 \cdot 5^3 = 5^4 \)[/tex]
3. [tex]\( 5^6 \cdot 5^{-4} = 5^2 \)[/tex]
4. [tex]\( 5^{-4} \cdot 5^4 \cdot 5^0 = 1 \)[/tex]
5. [tex]\( 5^7 \cdot 5^3 = 5^{10} \)[/tex]
6. [tex]\( 5^4 \)[/tex]
7. [tex]\( 5^0 = 1 \)[/tex]
8. [tex]\( \frac{1}{5^6} \)[/tex]
9. [tex]\( 5^2 \)[/tex]
10. [tex]\( 5^0 = 1 \)[/tex]
Each expression is simplified to reflect the rules of exponents.
