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Chapter 3: Problem 117

Simplify. $$-2^{6}$$

Short Answer

Expert verified
-64

Step by step solution

01

Understand the Expression

The expression is \(-2^{6}\). It is crucial to note that the negative sign is outside the exponentiation.
02

Evaluate the Exponentiation

Focus on calculating \(2^{6}\) first. This means multiplying 2 by itself 6 times: 2 \(\cdot\) 2 \(\cdot\) 2 \(\cdot\) 2 \(\cdot\) 2 \(\cdot\) 2 = 64.
03

Apply the Negative Sign

Once you have computed \(2^{6}\), apply the negative sign from the original expression. Thus, \(-2^{6} = -64\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Negative Sign
In mathematics, the placement of the negative sign is very important. In the expression \(-2^{6}\), the negative sign is outside the exponentiation. This means that only the base, 2, is raised to the power of 6. The negative sign is then applied to the result of this exponentiation.

It's a common error to think that the entire expression, including the negative sign, is raised to the power, but that is incorrect. In symbols, \((-2)^{6}\) would include the negative sign in the exponentiation process, but \(-2^{6}\) does not.
The Process of Simplification
Simplification means making an expression easier to understand or solve. When simplifying \(-2^{6}\), we start by solving the exponentiation part first: \(2^{6}\).

  • Step 1: Focus on the base and exponent, which in this example are 2 and 6.
  • Step 2: Multiply 2 by itself 6 times.
This step-by-step approach ensures clarity and accuracy in mathematical problems.
After computing the exponentiation, we then apply any arithmetic operations, such as applying the negative sign in this case. This methodical way of breaking down the problem helps to avoid mistakes and makes tackling complex expressions manageable.
Powers of Numbers
Exponentiation involves raising a number (the base) to a certain power (the exponent). In our example, we calculate \(2^{6}\).

Here are the steps:
  • Begin with the base number, which is 2.
  • Multiply 2 by itself, 6 times: 2 \cdot\ 2 \cdot\ 2 \cdot\ 2 \cdot\ 2 \cdot\ 2.
  • The product of these multiplications is 64.
This means that \(2^{6} = 64\).
Understanding these basics of powers of numbers can help you solve any problem that involves exponentiation. They are foundational concepts in algebra and many other areas of mathematics.

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