Question

Express each of the following as a power of 2:

a: 2³ × 4 × 8
b: 1/2 × (2¹5)
c: (2⁵)⁶ ÷ 3
d: (1/2)^(-2)

Answer 1

To express a given mix of numbers and operations as a power of 2, convert all numbers to powers of 2, then use the rules of exponents to simplify. This includes adding exponents when multiplying the same base, multiplying exponents when raising a power, and understanding negative exponents as reciprocals.

Step-by-step explanation:

Expressing Numbers as a Power of 2

To express each given expression as a power of 2, we'll simplify each one step by step:

1. 2³ × 4 × 8: Since 4 equals 2² and 8 equals 2³, the expression can be written as 2³ × 2² × 2³. Adding the exponents together, we get 2³+2+3 which is 2¸.
2. 1/2 × (2¹⁵): Recognizing that 1/2 is 2⁻¹, this can be rewritten as 2⁻¹ × 2¹⁵. When multiplying powers of 2, we add the exponents, resulting in 2⁻¹+15 which is 2¹⁴.
3. (2⁵)⁶ ÷ 3: Raising a power to a power means multiplying the exponents, so this is 2³⁰. Since this is divided by 3, which is not a power of 2, we express only the power of 2 as 2³⁰.
4. (1/2)⁻²: A negative exponent indicates the reciprocal, so this is the same as (2/1)², which simplifies to 2².

Keep in mind that exponentiation follows specific rules such as multiplying bases with the same number which allows adding exponents, raising a power to another power which involves multiplying exponents, and dealing with negative exponents which indicates taking the reciprocal of the base.