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Chapter 4: Problem 40

Evaluate. Assume that all variables represent nonzero real numbers. $$ -8^{0}-k^{0} $$

Short Answer

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Step by step solution

01

Title - Understanding the Zero Exponent Rule

Recall that any nonzero number raised to the power of 0 is 1. This means that for any nonzero real numbers, Now, we apply this rule to the given expression:
02

Title - Apply the Zero Exponent Rule

First, evaluate -8^{0} The zero exponent rule tells us: -8^{0} = 1 Next, evaluate k^{0} Using the same rule: k^{0} = 1 Thus the given expression is simplified as follows:
03

Title - Simplify the Expression

Now combine the results from step 2 into the original expression: -8^{0} - k^{0}which simplifies to: 1 - 1 = 0

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Key Concepts

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

Zero Exponent Rule
The Zero Exponent Rule is a fundamental principle in algebra. It states that any nonzero number raised to the power of 0 equals 1. This can be expressed mathematically as \(a^{0} = 1\) for any \(a \e 0\). Understanding this rule is key to simplifying expressions involving exponents. Let's put this into practice with an example:

-8^{0} and k^{0}.

By the Zero Exponent Rule: \-8^{0} = 1\ and \k^{0} = 1\. Using these results simplifies the expression -8^{0} - k^{0} to 1 - 1, which equals 0.
Exponents
Exponents are a shorthand way of indicating repeated multiplication. For instance, \(a^{n}\) means that 'a' is multiplied by itself 'n' times. They can simplify mathematical calculations significantly. Here are key points about exponents:

  • The base is the number being multiplied.
  • The exponent tells you how many times to multiply the base by itself.
  • Examples include \2^{3} = 2 \cdot 2 \cdot 2 = 8\ and \5^{2} = 5 \cdot 5 = 25\.
Exponents follow specific rules like the Zero Exponent Rule, product of powers rule, and the power of a power rule. These rules help when simplifying expressions involving exponents.
Simplifying Expressions
Simplifying expressions involves breaking them down into their simplest form. We do this using algebraic rules. Key steps include:

  • Applying exponent rules.
  • Combining like terms.
  • Following the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Simplifying an expression like -8^{0} - k^{0} involves using the Zero Exponent Rule to evaluate each term separately and then combining the results. Here, -8^{0} equals 1 and k^{0} equals 1, simplifying the expression to 1 - 1, which is 0.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(5 a^{-1}\right)^{4}\left(a^{2}\right)^{-3} $$

A student subtracted \(\left(15 x^{2}+12\right)-\left(7 x^{2}-3\right)\) vertically as follows. $$\begin{array}{r}15 x^{2}+12 \\\7 x^{2}-3 \\\\\hline 8 x^{2}+9\end{array}$$ Give the correct difference.

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(-5 y^{3} z^{4}\right)^{2}\left(2 y z^{5}\right)^{-2}}{10\left(y^{4} z\right)^{3}\left(3 y^{3} z^{2}\right)^{-1}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{2^{2} y^{4}\left(y^{-3}\right)^{-1}}{2^{5} y^{-2}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(3 r^{2}\right)^{2} r^{-5}}{r^{-2} r^{3}}\left(2 r^{-6}\right)^{2} $$
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