Problem 6 - Algebra for College Students

Question

Simplify each numerical expression. \(\frac{1}{2^{-6}}\)

Step-by-Step Solution

Answer

The simplified expression is 64.

Step 1: Understand the expression

The given expression is \( \frac{1}{2^{-6}} \). It involves a negative exponent. We're going to simplify this expression by using properties of exponents.

Step 2: Apply the property of negative exponents

One of the exponent properties is that \( a^{-n} = \frac{1}{a^{n}} \). Applying this property, we can rewrite the expression as \( \frac{1}{\frac{1}{2^6}} \).

Step 3: Simplify the division

Dividing by a fraction is the same as multiplying by its reciprocal. Thus, \( \frac{1}{\frac{1}{2^6}} \) simplifies to \( 1 \times 2^6 \), which is \( 2^6 \).

Step 4: Calculate the power

Now, compute \( 2^6 \). This means multiplying 2 by itself 6 times: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \). Thus, \( 2^6 = 64 \).

Key Concepts

Negative Exponents

A negative exponent means the reciprocal of the base raised to the absolute value of the exponent. This concept can seem puzzling at first, but it's actually quite simple. Think of the negative sign as a way to "flip" the fraction. For example, if you have a base of 2 with an exponent of -6, written as \(2^{-6}\), you rewrite it as \(\frac{1}{2^6}\).

In the context of our problem, \(\frac{1}{2^{-6}}\) becomes \(\frac{1}{\frac{1}{2^6}}\). Doing so helps in simplifying expressions without having to deal with negative exponents directly.

• Negative exponents indicate a reciprocal.
• Rewrite as a fraction with a positive exponent.
• This gives clarity and makes further calculations easier.

Understanding negative exponents allows you to take control of seemingly complicated expressions and convert them into manageable positive exponents.

Exponent Properties

The properties of exponents form the rules that guide how we manipulate expressions involving exponents. They are foundational to simplifying expressions. When you apply the negative exponent property, \( a^{-n} = \frac{1}{a^{n}} \), you're using a fundamental rule that dictates that negative exponents just mean reciprocal of the base raised to the positive exponent.

Beyond just negating and reciprocating, there are several other properties, such as the product of powers \(a^m \times a^n = a^{m+n}\), and the power of a power \((a^m)^n = a^{m \times n}\).

• Use the properties to convert negative exponents into positive ones.
• Combine terms wisely using product or power rules.
• Always check for simplifications at each step.

These rules help simplify any expression quickly and accurately with minimal computation.

Numerical Expressions

Numerical expressions are essentially mathematical phrases combining numbers and operations, such as addition or multiplication. When facing a problem like \(\frac{1}{2^{-6}}\), breaking it down step-by-step makes things clearer.

In simpler terms, you first remove the negative exponent by interpreting it as a reciprocal. Next, handle any arithmetic like division or multiplication.Converting the expression after applying properties of exponents, we simplify calculations by directly substituting \(2^6\) with its computed value, 64.

• Always start by identifying all operations involved.
• Apply operational properties like reciprocals before computing.
• Finally, resolve the expression by computing powers and performing arithmetic.

Mastering numerical expressions includes understanding and applying the right properties to simplify even complex-looking problems easily.