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Chapter 4: Problem 22

\((-5 t)^{0} \quad(t \neq 0)\)

Short Answer

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1

Step by step solution

01

Understand the Zero Exponent Rule

Any non-zero number raised to the power of 0 is equal to 1. This is known as the zero exponent rule.
02

Apply the Zero Exponent Rule

Since the exponent is 0, regardless of the base (-5t), as long as (t eq 0), the expression ((-5t)^0) simplifies directly to 1.
03

Final Answer

((-5t)^0) = 1

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Key Concepts

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

Exponent Rules
Exponent rules are guidelines that simplify working with powers. Powers are useful in algebraic expressions when numbers are multiplied repeatedly. The zero exponent rule is one such rule, stating any non-zero number raised to the power of zero equals 1. For example, \((-5t)^{0}\) simplifies to 1 immediately. Let's review main exponent rules:
  • Product of Powers Rule: \((a^m \times a^n = a^{m+n})\)
  • Power of a Power Rule: \((a^{m})^n = a^{m \times n}\)
  • Power of a Product Rule: \((a \times b)^n = a^n \times b^n\)
  • Quotient of Powers Rule: \((a^m / a^n = a^{m-n}, a eq 0)\)
These rules are foundational in algebra for simplifying expressions and solving equations.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. A typical algebraic expression might look like \((3x^2 - 4x + 7)\). Here:
  • Variables: letters like \(x\) and \(t\) that represent unknown values.
  • Constants: specific numbers like \(3\), \(-4\), and \(7\).
  • Coefficients: numbers multiplied by variables, such as \((3\) in \((3x^2)\).
  • Operations: addition, subtraction, multiplication, and division.
Expressions can be simplified using exponent rules. For example, in our exercise, we have an expression \((-5t)^0\). Given the zero exponent rule, this simplifies to 1, making it easier to work with.
Exponentiation
Exponentiation involves raising a number (the base) to a power (the exponent). The base is the number being multiplied by itself, and the exponent tells us how many times to multiply the base. For instance, in the expression \((3^4)\), \(3\) is the base, and \(4\) is the exponent, meaning \(3 \times 3 \times 3 \times 3\).
Some important points about exponentiation include:
  • Positive Exponents: Tell how many times to multiply the base by itself.
  • Negative Exponents: Indicate reciprocal multiplication. For example, \(a^{-n} = 1/a^n\).
  • Zero Exponent: Any non-zero number to the power of zero is 1, as explained in our exercise with \((-5t)^0 = 1\).
Understanding these concepts is crucial for mastering algebra and other areas of mathematics.

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Most popular questions from this chapter

If it costs \(\$ 15\) plus \(\$ 2\) per day to rent a chain saw, the binomial $$2 x+15$$ gives the cost in dollars to rent the chain saw for \(x\) days. Evaluate this binomial for \(x=6\). Use the result to fill in the blanks: If the saw is rented for_________ days, then the cost is _________ dollars.

$$ \begin{aligned} &y=-x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Perform each indicated operation. \(\left[\left(3 x^{2}-2 x+7\right)-\left(4 x^{2}+2 x-3\right)\right]-\left[\left(9 x^{2}+4 x-6\right)+\left(-4 x^{2}+4 x+4\right)\right]\)

Find each product. $$ (5-3 x)(4+x) $$

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