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Chapter 10: Problem 235

In the following exercises, simplify. $$-7^{0}$$

Short Answer

Expert verified
The simplified expression is \(-1\).

Step by step solution

01

Understand exponent rules

Recall that any number raised to the power of zero is equal to 1. This is known as the zero exponent rule. Mathematically, for any non-zero number a, \(a^0 = 1\).
02

Apply the zero exponent rule

Apply the zero exponent rule to the given expression. Here, the expression is \(-7^0\). Using the rule, replace \(7^0\) with 1.
03

Simplify the expression

Since \(7^0 = 1\), the original expression \(-7^0\) simplifies to \(-1\).

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Key Concepts

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

zero exponent rule
One of the fundamental rules in the study of exponents is the zero exponent rule. This rule states that any non-zero number raised to the power of zero equals 1. In mathematical terms, if we have a non-zero number represented as 'a', then the rule can be written as \(a^0 = 1\). This rule is essential to understand and is frequently used in various mathematical problems. The intuition behind this rule is related to decreasing the exponent: as we decrease the exponent by one, we divide the base by itself when the exponent hits zero, leaving us with 1. Hence, \(7^0\) simplifies to 1. Knowing and remembering this rule helps in simplifying more complex expressions involving exponents.
exponents
Exponents are a way to express repeated multiplication of a number by itself. For instance, in the expression \(3^4\), the number 3 is the base, and the number 4 is the exponent. This means 3 is multiplied by itself 4 times: \(3 \times 3 \times 3 \times 3\). Exponents follow specific rules that help simplify expressions including:
  • Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power Rule: \((a^m)^n = a^{mn}\)
  • Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\) for \(a eq 0\)
These rules are crucial for combining and breaking down expressions involving exponents.
negative numbers
Negative numbers are numbers less than zero, represented with a minus sign such as -1, -2, -3, etc. When dealing with exponents and negative numbers, we need to be mindful of the signs. For example, in the exercise \(-7^0\), we can see two parts: the negative sign and the exponentiation. According to the zero exponent rule, \(7^0\) simplifies to 1. Thus, \(-7^0\) becomes \(-1\). This indicates that while exponents have specific rules, the ordinary rules of arithmetic, such as handling negative signs, still apply. Always carefully look at the context where negative numbers interact with other operations.

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