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Chapter 6: Problem 93

Simplify each expression with exponents. (a) \((-2)^{6}\) (b) \(-2^{6}\)

Short Answer

Expert verified
(a) \((-2)^{6} = 64\). (b) \-2^{6} = -64\.

Step by step solution

01

Simplify \((-2)^{6}\)

The expression \((-2)^{6}\) means that \(-2\) is raised to the power of 6. This can be computed as: \-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2\. Since the exponent is even, the result will be positive. Therefore, \((-2)^{6} = 64\).
02

Simplify \-2^{6}\

The expression \(-2^{6}\) means that the negative sign is not included in the exponentiation. This is equivalent to \-1 \cdot (2^{6})\. First compute \(2^{6} = 64\), and then multiply by \-1\. Therefore, \-2^{6} = -64\.

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Key Concepts

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

Exponentiation
Exponentiation is a basic algebraic operation that involves raising a base to the power of an exponent. It is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent. This means multiplying the base by itself as many times as indicated by the exponent. For example, in the expression \((-2)^6\), the base is \(-2\) and the exponent is 6. So, you compute \(-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2\). Since there are an even number of negative signs, they pair up to make positive numbers, resulting in a positive outcome: 64.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest form. This involves calculating all operations such as multiplication, division, addition, and subtraction, along with working through any exponents. For example, simplifying the expression \(-2^6\) involves recognizing that the negative sign is not inside the parentheses, so it does not get raised to the power with the 2. This becomes \(-1 \cdot 2^6\). Calculate \(2^6 = 64\), then multiply by -1 to get -64.
Simplification follows the order of operations rules (PEMDAS/BODMAS) where you handle Parentheses and Exponents first.
Negative Bases
Negative bases in exponents can be tricky because the outcome depends on whether the exponent is even or odd. If the negative base is in parentheses, like \((-2)^6\), each step of multiplication involves a negative number. When done an even number of times, the negatives cancel out and leave a positive result: 64. However, if the base is not in parentheses, like \(-2^6\), the negative sign is not part of the exponentiation. Instead, you are multiplying -1 by the result of \(2^6\), which is 64. Thus, the result is -64.
Understanding this difference is key to mastering algebraic exponentiation with negative bases.

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