Sure, let's break down each part of the question and solve it step by step.
### 3. Express [tex]\(4^{-0}\)[/tex] as a power
Any number raised to the power of 0 is equal to 1. Therefore,
[tex]\[ 4^{-0} = 4^0 = 1 \][/tex]
### 4. Simplify and write in exponential form
(i) Simplify [tex]\(\left(2^5 \div 2^8\right)^5 \times 2^{-5}\)[/tex]
To simplify this expression, we can use the properties of exponents. Here are the steps:
1. Consider the inner expression [tex]\(\left(2^5 \div 2^8\right)\)[/tex]:
[tex]\[ \frac{2^5}{2^8} = 2^{5-8} = 2^{-3} \][/tex]
2. Raise the result from step 1 to the power of 5:
[tex]\[ (2^{-3})^5 \][/tex]
3. Use the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (2^{-3})^5 = 2^{-3 \cdot 5} = 2^{-15} \][/tex]
4. Multiply the result by [tex]\(2^{-5}\)[/tex]:
[tex]\[ 2^{-15} \times 2^{-5} \][/tex]
5. Use the product of powers property [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[ 2^{-15 + (-5)} = 2^{-20} \][/tex]
Therefore, the simplified form of the expression [tex]\(\left(2^5 \div 2^8\right)^5 \times 2^{-5}\)[/tex] is:
[tex]\[ 2^{-20} \][/tex]
### Summary
3. [tex]\( 4^{-0} = 1 \)[/tex]
4. [tex]\((\left(2^5 \div 2^8\right)^5 \times 2^{-5}\)[/tex] simplifies to [tex]\(2^{-20}\)[/tex]
