/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Simplify the given expressions. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Simplify the given expressions. Express results with positive exponents only. $$x^{3} x^{4}$$

Short Answer

Expert verified
The simplified expression is \( x^7 \).

Step by step solution

01

Identify the Expression

We are given the expression to simplify: \[ x^3 x^4 \] Our goal is to simplify this such that all exponents are positive.
02

Apply the Product of Powers Rule

The product of powers rule states that when multiplying like bases, you add the exponents. For base \(x\), the rule can be expressed as: \[ x^a \, x^b = x^{a+b} \] In this problem, apply the rule to combine \(x^3\) and \(x^4\) by adding their exponents: \[ x^{3+4} \]
03

Simplify the Exponential Expression

Add the exponents together to simplify the expression: \[ x^{3+4} = x^7 \] This is the simplified expression with positive exponents only.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Product of Powers Rule
When we talk about the Product of Powers Rule, it's like having a little shortcut in math when working with exponents. This rule tells us that if you are multiplying two numbers or terms that have the same base, you can simply add their exponents together. For example:
  • When you have something like \( x^a \times x^b \) where \( x \) is the common base, you can simplify this to \( x^{a+b} \).
  • This means if you multiply \( x^3 \times x^4 \), you can just add the 3 and 4 to get \( x^7 \).
It's a neat way to make complex multiplication problems a lot simpler. Just remember, this only works when the bases are the same. So using the Product of Powers Rule can save us time and make calculations more manageable.
Exponents
Exponents are a way to express repeated multiplication of the same number. They're like shorthand in math. Instead of writing out long multiplication, you can use an exponent to tell you how many times to multiply a number by itself.
  • For example, \( x^3 \) means you multiply \( x \) by itself 3 times: \( x \times x \times x \).
  • It's a little number written above and to the right of a base number, indicating the power of that base.
Simply put, exponents can simplify operations by reducing the need to write out long strings of numbers. They are widely used in algebra and beyond, making them a foundational concept in math. Understanding how to manipulate exponents, like through addition or subtraction, is crucial when you're working on more advanced problems.
Simplifying Expressions
Simplifying expressions involves breaking down complex expressions into simpler forms that are easier to manage and interpret. When you simplify, you aim to express things with as few operations or steps as possible.
  • With the expression \( x^3 \times x^4 \), we see how applying the Product of Powers Rule instantly reduces the complexity to \( x^7 \).
  • This is a form of simplification because it converts a lengthy operation into a single, easy-to-understand power term.
The goal is to make expressions more readable and easier to work with, especially when dealing with large algebraic equations or when trying to insert these expressions into larger equations. Remember, simplifying doesn't change the value of the expression, just the way we express that value.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Perform the indicated divisions. Find \(k\) such that the remainder of the division $$\left(6 x^{3}-x^{2}-14 x+k\right) \div(3 x-2) \text { is }-2.$$

Solve the given problems. For what values of \(x\) is (a) \(x>\sqrt{x},\) (b) \(x=\sqrt{x}\), and (c) \(x<\sqrt{x} ?\)

Perform the indicated divisions. In analyzing the displacement of a certain valve, the expression \(\frac{s^{2}-2 s-2}{s^{4}+4}\) is used. Find the reciprocal of this expression and then perform the indicated division.

Perform the indicated divisions. In analyzing a rectangular computer image, the area and width of the image vary with time such that the length is given by the expression \(\frac{2 t^{3}+94 t^{2}-290 t+500}{2 t+100} .\) By performing the indicated division, find the expression for the length.

Solve the given equations. $$1.5 x-0.3(x-4)=6$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.