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Chapter 5: Problem 45

Simplify each expression. $$ (-6 t)^{0} $$

Short Answer

Expert verified
The expression simplifies to 1.

Step by step solution

01

Understanding the Zero Exponent Rule

The zero exponent rule states that any non-zero base raised to the power of zero is equal to 1. In other words, if \( a eq 0 \), then \( a^0 = 1 \).
02

Apply the Zero Exponent Rule

In the expression \((-6t)^0\), the entire expression is the base raised to the power of zero. By applying the zero exponent rule, since the base is non-zero, we have \((-6t)^0 = 1\).

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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 may sound complex, but it's quite straightforward and incredibly useful in algebra. Whenever you see an expression where a number or variable is raised to the power of zero, this rule comes into play. Here鈥檚 the magic:
  • If any non-zero number or variable, let's call it \(a\), is raised to the 0 power, it equals 1.
So the mathematical expression \(a^0 = 1\) holds true as long as \(a\) is not zero. For example, \((-6t)^0 = 1\). The logic is that an "empty product" of any number of factors is considered to be 1, a concept which is fundamental in maintaining consistency across various mathematical operations.
Exponents
Exponents, sometimes called powers or indices, are a way to express repeated multiplication of a number by itself. The expression \(b^n\) represents \(b\) being multiplied by itself \(n\) times. Knowing how to handle exponents is crucial for simplifying expressions and solving equations.Here are some key points:
  • The base \(b\) is the number or variable that is multiplied.
  • The exponent \(n\) tells you how many times to multiply the base by itself.
Think of the exponent as a "shortcut code" for multiplication. For example, \(5^3\) means \(5 \times 5 \times 5\). This concept allows us to express large numbers in a compact form and solve algebraic problems more efficiently.
Simplification of Expressions
Simplifying expressions is an essential skill in algebra that involves rewriting them in a more concise, easier-to-understand form. This process often requires the application of various algebraic rules and properties, including exponent rules. When simplifying, you might:
  • Combine like terms, which involves adding or subtracting terms that have the same variables raised to the same power.
  • Apply exponent rules, such as the zero exponent rule, to make the expression simpler.
For instance, using the zero exponent rule, the expression \((-6t)^0\) simplifies directly to 1. Simplification makes solving equations easier and helps to reveal the underlying relationships between variables.

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Most popular questions from this chapter

Factor. Assume all variables represent natural numbers. $$ 4 x^{2 n}-9 y^{2 n} $$

On a test, when asked to find \((x-y)^{2},\) a student answered \(x^{2}-y^{2} .\) What error did the student make?

Multiply. Assume \(n\) is a natural number. $$ a^{2 n}\left(a^{n}+a^{2 n}\right) $$

Factor each expression. $$ (c-d)^{3}+216 $$

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. \(a^{6}-b^{3}\) b. \(a^{6}+b^{3}\)
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