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Put the simplified expressions in order on the number line from least to greatest.

1. 1/4^-2
2. 2^6 -2^4/2^2
3. (1/2)^-3
4. (-2)^-8 * (-2)^11

1 answer

Let's simplify each expression step by step:

  1. \( \frac{1}{4^{-2}} \): \[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \] Therefore: \[ \frac{1}{4^{-2}} = 4^2 = 16 \]

  2. \( 2^6 - \frac{2^4}{2^2} \): First, simplify \( \frac{2^4}{2^2} \): \[ \frac{2^4}{2^2} = 2^{4 - 2} = 2^2 = 4 \] Then, calculate \( 2^6 - 4 \): \[ 2^6 = 64 \quad \text{so} \quad 64 - 4 = 60 \]

  3. \( \left( \frac{1}{2} \right)^{-3} \): \[ \left( \frac{1}{2} \right)^{-3} = 2^3 = 8 \]

  4. \( (-2)^{-8} * (-2)^{11} \): When multiplying powers with the same base, we add the exponents: \[ (-2)^{-8} * (-2)^{11} = (-2)^{-8 + 11} = (-2)^3 \] Calculate \( (-2)^3 \): \[ (-2)^3 = -8 \]

Now we have simplified all expressions:

  1. \( 16 \)
  2. \( 60 \)
  3. \( 8 \)
  4. \( -8 \)

Putting these values in order from least to greatest:

  • \( -8 \) (from expression 4)
  • \( 8 \) (from expression 3)
  • \( 16 \) (from expression 1)
  • \( 60 \) (from expression 2)

Therefore, the order is: \[ \text{4, 3, 1, 2} \]