Problem 10 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(-2^{6} \cdot 2^{2}\)

Step-by-Step Solution

Answer

The simplified value of \(-2^{6} \cdot 2^{2}\) is 256.

Step 1: Understand the Problem

We need to simplify the expression \(-2^{6} \cdot 2^{2}\). This involves simplifying the product of two powers with the same base.

Step 2: Apply the Exponent Rule

When multiplying powers with the same base, you can add their exponents. The rule is: \(a^{m} \cdot a^{n} = a^{m+n}\). Here, the base is 2. Therefore, \(2^{6} \cdot 2^{2} = 2^{6+2}\).

Step 3: Simplify the Expression

Add the exponents to obtain \(2^{8}\). We now have the expression simplified to a single exponent.

Step 4: Evaluate the Expression

Calculate \(2^{8}\). Since \(2^{8} = 256\), the simplified value of the expression is 256.

Key Concepts

Simplifying Expressions

Simplifying expressions may sometimes appear daunting, but it becomes manageable with some simple guidelines. When dealing with expressions that involve powers or exponents, breaking them down into more straightforward parts helps drastically.
Begin by observing the operation connecting each part of the expression. In cases involving multiplication of similar bases, focus on those bases alone. Once you identify them, apply the relevant exponent rules to simplify. After simplification, consolidate the expression into one coherent piece if possible, which will often involve reducing it to an expression with a single base and exponent.
Also, remember to tackle negative signs with attention. If any part of the expression includes a negative sign, consider it as separate from the base when combining exponents. In our example, \(-2^{6} \cdot 2^{2}\), the negative sign stands apart from the base and exponent simplifications used later in the exercise.

Multiplying Powers

Multiplying powers efficiently is a crucial tool in algebra. A common scenario involves products of powers which share the same base. For such operations, simply add the exponents together to find the solution.
For example, consider the expression \(2^{6} \cdot 2^{2}\). Here, the base, 2, is consistent in both terms. By applying the exponent rule \(a^{m} \cdot a^{n} = a^{m+n}\), you can easily transform \(2^{6} \cdot 2^{2}\) into \(2^{8}\).
It's important to ensure the bases are the same before combining exponents. If they differ, this rule will not apply, and another approach will be necessary. Understanding this basic rule of multiplying powers allows for a more fluent approach to handling complex expressions, making them much easier to manage.

Evaluating Exponents

Evaluating exponents means figuring out what the power of a number equals. Once an expression is simplified to one term with a power, the next step is evaluation.
Take, for instance, the expression \(2^{8}\). This means you'll multiply the base, which is 2, by itself repeatedly, 8 times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\). Through this multiplication, you find the value equals 256.
It's incredibly vital to practice the evaluation of simpler exponents manually, as this builds a stronger understanding and strengthens mental math skills. However, for larger exponents, using a calculator can help save time and ensure accuracy. Being able to evaluate exponents confidently is an essential skill in simplifying expressions and solving math problems efficiently.