Question

Simplify each expression. \(\left(2^{-1}\right)^{-3}\)

Short answer

8

Step-by-step solution

Understand negative exponents

Negative exponents mean reciprocal. For any nonzero number a, the expression \(a^{-n} = \frac{1}{a^n}\).

Apply the power of a power rule

The power of a power rule states that \((a^m)^n = a^{mn}\). Apply this rule to the expression \( \big(2^{-1} \big)^{-3} \). This gives us \ 2^{-1 \times -3} = 2^3 \.

Calculate the exponent

Simplify the exponent step: \(2^3 = 2 \times 2 \times 2 = 8 \).

Key concepts

Exponent Rules

Understanding exponent rules is crucial when working with exponential expressions. These rules help us manipulate and simplify different forms of expressions involving exponents. One important rule is the rule for negative exponents. When you have a negative exponent like in the expression \(a^{-n}\), it translates to the reciprocal of the base raised to the positive exponent: \(a^{-n} = \frac{1}{a^n}\).
Another useful rule is the power of a power rule which states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\).
Mastering these rules makes it easier to handle more complex problems involving exponents.

Reciprocal

The concept of a reciprocal is vital in simplifying expressions involving negative exponents. The reciprocal of a number is essentially flipping it. For instance, the reciprocal of a number \(a\) is \(\frac{1}{a}\).
When we encounter negative exponents, such as \(a^{-n}\), it indicates that we should take the reciprocal of the base raised to the positive exponent: \(a^{-n} = \frac{1}{a^n}\).
For example, \(2^{-1}\) means taking the reciprocal of 2, which gives us \(\frac{1}{2}\). Understanding reciprocals and their relationship with negative exponents helps simplify expressions easily.

Power of a Power Rule

The power of a power rule helps us simplify expressions where an exponent is raised to another exponent. According to this rule, \((a^m)^n = a^{mn}\).
Let's apply this rule to the expression \(\big(2^{-1}\big)^{-3}\). First, recognize that we need to multiply the exponents: \((-1)\times(-3)\). Using the power of a power rule simplifies this to \(2^{3}\). Now, the expression is much simpler to evaluate.
This rule reduces complex nested exponents, making calculations straightforward and easier to manage.

Simplifying Expressions

Simplifying expressions involves breaking down complex problems into simpler, easy-to-handle parts. With the expression \(\big(2^{-1}\big)^{-3}\), the first step is recognizing and applying the appropriate exponent rule. Start by applying the power of a power rule, which simplifies it to \(2^{3}\).
Next, calculate the exponent to get the final value. Here, \(2^{3}\) means \(2 \times 2 \times 2 = 8\).
Simplifying expressions requires understanding the right rules and applying them step by step to reach a clear and accurate result.