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Chapter 7: Problem 3

Simplify: \(t^{8} \cdot t^{-4}\)

Short Answer

Expert verified
The simplified expression is \(t^4\).

Step by step solution

01

Understand the Problem

You are given the expression \(t^8 \cdot t^{-4}\) and need to simplify it. This involves combining the exponents based on the multiplication property of exponents.
02

Apply the Exponent Rule

The rule for multiplying powers of the same base is to add the exponents: \(a^m \times a^n = a^{m+n}\). Here, you will add \(8\) and \(-4\).
03

Perform the Addition

Add the exponents \(8\) and \(-4\): \(8 + (-4) = 4\).
04

Write the Simplified Expression

Using the result from the addition, the expression simplifies to \(t^4\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions involves reducing them to their most basic form without changing their value. It's like cleaning up a messy room, organizing everything neatly. In math, simplifying helps us handle expressions more easily.

When you simplify, you usually combine like terms, use rules and properties to make things simpler, and eliminate any unnecessary parts. For expressions with exponents, applying exponent rules can often make the process more efficient. In the case of multiplying terms with the same base, you rely on the multiplication property of exponents. This allows you to add the exponents together, streamlining the expression down to one term.
Negative Exponents
Negative exponents might seem tricky at first, but they're actually just a different way to represent fractions. Instead of thinking negatively, think about flipping the base.
- A negative exponent - What does this mean? Simply put, when you see an exponent that is negative, you can move the base to the other part of a fraction. For example, - This rule applies when simplifying expressions that include both positive and negative exponents, as it helps bring everything together neatly.
Multiplication Property of Exponents
The multiplication property of exponents is a handy rule that allows you to simplify expressions with the same base. This rule states: - when you multiply numbers with the same base. This means you simply add the exponents together to combine them.
For instance, if you have an expression like - The rule allows us to directly add up the exponents because the base remains the same. Thus, the expression can be quickly simplified by just calculating the sum of the exponents and raising the base to this new power. It's a great way to make expressions neater and more manageable, especially in algebra and higher-level math courses.

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Most popular questions from this chapter

Simplify: \(\left(\frac{5}{4}\right)^{-3}\).

Solve for \(x: 1-\frac{6}{x}=-\frac{8}{x^{2}}\)

Given that \(y\) is proportional to \(x\) and that \(y=21\) when \(x=9\), determine the value of \(y\) when \(x=27\).

Simplify: \(\left(-5 x^{8} y^{-2}\right)\left(-2 x^{-6} y^{-1}\right)\).

Suppose that the price per person for a tour is inversely proportional to the number of people who sign up for the tour. If 8 people sign up, the price per person is \(\$ 70\). What will be the price per person if 20 people sign up?
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