Question

Evaluate:

(4^–5)² ÷ 8^–6

Answer 1

Let’s evaluate the expression step by step:

We are asked to evaluate:

(4^–5)² ÷ 8^–6

Step 1: Simplify (4^–5)²

Use the property of exponents: (a^m)ⁿ = a^{m × n}

(4^–5)²

= 4^{–5 × 2}

= 4^–10 

Step 2: Express 4 and 8 as powers of 2

We know that 4 = 2² and 8 = 2³.

Now, let’s rewrite 4^–10 and 8^–6 using these base –2 representations.

4^–10 = (2²)^–10

= 2^{2 × –10}

= 2^–20

8^–6 = (2³)^–6

= 2^{3 × –6}

= 2^–18

Step 3: Simplify the expression

Now, the expression becomes `2^-20/2^-18`

Using the property of exponents `a^m/a^n = a^(m - n)`, we can simplify this:

2^{–20 – (–18)} 

= 2^{–20 + 18}

= 2^–2

Step 4: Final answer

`2^-2 = 1/2^2 = 1/4`

Thus, the value of the expression is `1/4`