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Chapter 2: Problem 56

Simplify. $$-1^{5}$$

Short Answer

Expert verified
The simplified form is -1.

Step by step solution

01

Understand the Expression

The expression to simplify is $$-1^{5}$$ This means $$- (1^5)$$ since the exponentiation is done before the negative sign applies.
02

Calculate the Exponentiation

Calculate $$1^{5}$$ Raising 1 to any power always results in 1. Therefore, $$1^{5} = 1$$.
03

Apply the Negative Sign

Now apply the negative sign to the result from Step 2. $$- (1) = -1$$.

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Key Concepts

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

negative exponents
A negative exponent tells you to divide 1 by the base raised to the positive exponent. For example, if you have an expression like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\).

Breaking it down:
  • Start with the base, let's say 'a'.
  • The negative exponent indicates you are dealing with the reciprocal of the base raised to the positive exponent.

    Here's a quick example: \(2^{-3}\). This becomes \(\frac{1}{2^3}\). Simplifying the exponentiation, \(2^3 = 8\). Finally, \(\frac{1}{8}\).

    Handy tip: The negative sign in an exponent does not make the result negative. It simply indicates a reciprocal operation.
order of operations
The order of operations is a set of rules that determines the correct sequence to evaluate a mathematical expression. An easy way to remember this is with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Taking an example, let's simplify \(-1^5\). According to PEMDAS, we must handle the exponentiation before applying the negative sign.

Steps to follow:
  • Identify and solve any terms inside parentheses first.
  • Next, address any exponents or powers.
  • Then, perform multiplication and division in order from left to right.
  • Lastly, do addition and subtraction from left to right.

    For expression simplification, it's crucial to strictly adhere to these rules to avoid errors. In our example, solve \(1^5\) first to get 1, and then apply the negative sign: \( -1\).
powers of numbers
Powers of numbers, also known as exponents, indicate how many times a number (the base) is multiplied by itself. For example, in \(2^4\), the base is 2 and the exponent is 4, meaning \(2 \times 2 \times 2 \times 2\ = 16\).

Some key points to remember:
  • Any number raised to the power of 0 is 1: \(a^0 = 1\), provided \(a eq 0\).
  • Any number raised to the power of 1 is the number itself: \(a^1 = a\).
  • Raising a number to the power of a positive integer involves multiplying the number by itself multiple times.

    Consider the expression \(1^5\): here, the number 1 is the base and 5 is the exponent. Raising 1 to any power, regardless of what the exponent is, will always result in 1. That鈥檚 because multiplying 1 by itself any number of times is still 1: \(1 \times 1 \times 1 \times 1 \times 1 = 1\).

    Therefore, to simplify \( -1^5 \), first compute \( 1^5 \) to get 1, and then apply the negative sign to get \(-1\).

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Most popular questions from this chapter

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