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Chapter 15: Problem 37

Simplify. $$ (-1)^{4}+2^{3} $$

Short Answer

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Step by step solution

01

- Simplify \((-1)^{4}\)

Calculate \((-1)^{4}\). When raising \-1\ to any even power, the result is always 1 because the negative signs cancel each other out. So, \((-1)^{4} = 1\).
02

- Simplify \(2^{3}\)

Calculate \(2^{3}\). Raising 2 to the power of 3 means multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\).
03

- Add the Results

Add the results from Step 1 and Step 2. So, \(1 + 8 = 9\).

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Key Concepts

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

Power of a Number
Exponentiation refers to the operation where a number (known as the base) is raised to the power of an exponent. The exponent tells us how many times to multiply the base by itself.

For example, in the expression \(2^3\), the base is 2 and the exponent is 3. This means we multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

It's important to note that exponentiation is different from regular multiplication or addition. Each step of exponentiation involves multiplying the base again.

This principle can be applied to any number since the exponent dictates the number of times the base is used as a factor.
Negative Base with Even Exponent
When dealing with a negative base raised to an even exponent, the result is always positive. This occurs because each pair of negative signs cancels out.

For example, consider \[(-1)^4\]. This can be broken down as \[(-1) \times (-1) \times (-1) \times (-1)\]. The first pair of \[(-1) \times (-1)\] results in 1, and the second pair also results in 1. Thus, \[1 \times 1 = 1\].

So, when we raised \(-1\) to the power of 4, we ended up with \1\. The same principle applies regardless of the negative base or even exponent you use. For example, \((-3)^{2} = 9\) and \((-5)^{4} = 625\) because the repetitive multiplication steps cancel out the negative signs.
Simplification of Exponents
Simplifying expressions involving exponents requires using the rules of exponentiation effectively. Start by calculating the power of each base.

Take the original exercise \[(-1)^4 + 2^3\] as an example. We already know that \((-1)^4 \) equals 1.

Next, compute \(2^3\), which is \(2 \times 2 \times 2 = 8\).

After calculating the individual exponentiations, the final step is to simplify the expression by adding the results. In this case, \[1 + 8 = 9\].

In summary, the process involves:
  • Calculating each power separately
  • Performing the arithmetic operations indicated (like addition, subtraction, etc.)
Simplifying expressions helps to understand and solve them more efficiently.

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