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Chapter 4: Problem 18

\(-(-13)^{0}\)

Short Answer

Expert verified
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Step by step solution

01

Identify the Exponent

Recognize that the expression \(-(-13)^0\) contains an exponent. Any number raised to the power of 0 is 1.
02

Simplify the Exponent

Simplify \(-13^0\). Like previously mentioned, any number to the power of 0 is 1. So, \(-13^0 = 1\).
03

Apply the Negative Sign

Take the negative sign outside of the parentheses into account. Therefore, \(-(1) = -1\).

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Key Concepts

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

Exponent Rules
When working with exponents, it's crucial to understand some fundamental rules. Exponents denote repeated multiplication. For example, in the expression \(a^n\), \(a\) is called the base and \(n\) is the exponent indicating how many times the base is multiplied by itself.

Here are a few basic exponent rules:

  • Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\)
  • Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n}\)
  • Power of a Power Rule: \((a^m)^n = a^{mn}\)
  • Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
  • Power of a Quotient Rule: \((\frac{a}{b})^n = \frac{a^n}{b^n}\)

Remember, these rules make it easier to simplify expressions involving exponents.

In our original problem, understanding these rules helps us identify that any base raised to the zero exponent equals 1.
Zero Exponent
The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. Mathematically, we express this as \(a^0 = 1\), where \ a \eq 0\.

This rule holds true regardless of the base. For instance:

  • \(5^0 = 1\)
  • \((-9)^0 = 1\)
  • \(1000^0 = 1\)

In our exercise, we have the expression \(-(-13)^0\).\

First, we handle the exponent part: \(-13^0 = 1\).
Then, don't forget the negative sign outside the parentheses. This transforms the expression to \(-1\).
Negative Numbers
Negative numbers can sometimes be tricky, especially when they interact with other mathematical elements like exponents. It鈥檚 crucial to understand the distinction between negative numbers raised to an exponent and negative numbers serving as the base.

For instance:
  • \textcolor{gray}{Negative Base With Even Exponent}\textcolor{black}{:} \((-2)^4 = 16\)
  • \textcolor{gray}{Negative Base With Odd Exponent}\textcolor{black}{:} \((-3)^3 = -27\)
  • \textcolor{gray}{Negative Number Raised to Zero}\textcolor{black}{:} \((-14)^0 = 1\)

Be mindful of parentheses. For example:

\((-5)^2 = 25\) is different from \ -5^2 = -25\.

In the original exercise, we had \(-13^0 = 1\). When including the negative sign from outside the parentheses, \(-(-13)^0 = -1\).

This distinction helps avoid mistakes and ensures the correct understanding of expressions.

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Most popular questions from this chapter

Find each product. $$ 2 x^{2}(2 x-5)(x+3) $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (3 m-1)^{2} $$

Find each product. $$ -3 r(r-1)\left(r^{2}+r+1\right) $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5 k^{2}\left(k^{3}-3\right)\left(k^{2}-k+4\right) $$

Perform each indicated operation. \(\left[\left(9 b^{3}-4 b^{2}+3 b+2\right)-\left(-2 b^{3}-3 b^{2}+b\right)\right]-\left(8 b^{3}+6 b+4\right)\)
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