Evaluate each exponential expression in Exercises 1–22. $2^3/2^7$

Verified step by step guidance

1. Identify the given expression: \$2^3 / 2^7$.
2. Recall the property of exponents for division: when dividing powers with the same base, subtract the exponents. This means \(\frac{a^m}{a^n} = a^{m-n}\).
3. Apply the property to the expression: \$2^3 / 2^7 = 2^{3-7}$.
4. Simplify the exponent: \$3 - 7 = -4\(, so the expression becomes \)2^{-4}$.
5. Recognize that a negative exponent means the reciprocal: \$2^{-4} = \frac{1}{2^4}\(. You can leave the answer in this form or evaluate \)2^4$ if needed.

Key Concepts

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers. Key rules include multiplying powers with the same base by adding exponents, dividing by subtracting exponents, and raising a power to another power by multiplying exponents. These rules help simplify expressions like 2^3 / 2^7.

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying exponent rules to rewrite the expression in a simpler form. For division with the same base, subtract the exponent in the denominator from the exponent in the numerator. For example, 2^3 / 2^7 simplifies to 2^(3-7) = 2^(-4).

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For instance, a^(-n) = 1 / a^n. This concept is essential when simplifying expressions like 2^(-4), which equals 1 / 2^4 = 1/16.