Problem 7 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 2^{-4} \cdot 2^{5} $$

Step-by-Step Solution

Answer

The evaluated expression is 2.

Step 1: Define the expression

The expression to be evaluated is \(2^{-4} \cdot 2^{5}\)

Step 2: Apply the rules of exponents

When multiplying expressions with the same base, we add the exponents together. This operation gives us \(2^{-4+5}\).

Step 3: Simplify the exponent

Simplify the exponent by performing the addition operation, which gives us \(2^{1}\).

Step 4: Evaluate the expression

Since \(2^{1} = 2\), the evaluated expression is 2.

Key Concepts

Negative Exponents

Negative exponents can be a bit confusing at first, but they are actually quite straightforward once you know the rules. When you see an exponent with a negative sign, such as \(2^{-4}\), it means the reciprocal of the base raised to the positive version of the exponent. In other words, \(a^{-n}\) equals \(\frac{1}{a^n}\). So for \(2^{-4}\), it becomes \(\frac{1}{2^4}\) or \(\frac{1}{16}\). This concept transforms potentially intimidating negative exponents into simple fractions, making it easier to dive into calculations.
Remember:

• Negative exponents indicate the reciprocal of a number.
• They turn powers into fractions.
• Transform \(a^{-n}\) to \(\frac{1}{a^n}\).

Understanding negative exponents allows you to handle them with confidence in expressions.

Multiplying Powers with the Same Base

One fun and powerful rule of exponents is how they interact when you multiply powers with the same base. For example, the expression \(2^{-4} \cdot 2^{5}\) showcases two exponents having the same base, which is 2 in this case.
Whenever you multiply two powers with the same base, what you do is simple: **add the exponents together**. Mathematically, the rule is expressed as \(a^m \cdot a^n = a^{m+n}\).
In our example:

• Base: 2
• Exponents: -4 and 5
• Summed Exponent: \(-4 + 5\)
• Result: \(2^{1}\)

This technique makes complex-looking expressions much easier to work with.

Simplifying Expressions

Simplifying expressions is the magic trick in math that turns complex calculations into simple answers, saving you time and effort. When you have multiple operations, especially involving exponents, following the basic rules to simplify can make a big difference.
Let's take the expression \(2^{-4} \cdot 2^{5}\) from our exercise:

• After using the exponent rules, we got \(2^{1}\).
• This simplification tells us that instead of long complex calculations, we simply have "2" as our answer, because \(2^1 = 2\).

Simplification is about extracting the core value from the bunch of numbers and operations. It's crucial for making math less of a challenge and more of a breeze in many situations. Focus on the steps—apply the right exponent rules, be aware of operations, and the math will usually work itself out.