/*! 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 78 Simplify. $$\left(x^{2} x^{3}\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 78

Simplify. $$\left(x^{2} x^{3}\right)^{5}$$

Short Answer

Expert verified
The simplified form of the expression is \(x^{25}\).

Step by step solution

01

Understand the problem

The expression \( (x^{2} x^{3})^{5} \) involves the multiplication of exponents, followed by raising to the power of \(5\). The goal is to simplify the entire expression into a single power of \(x\).
02

Multiply the exponents

First, we simplify \(x^{2} x^{3}\). Since the bases are the same, we can add the exponents, which yields \(x^{5}\). This gives us \((x^{5})^{5}\).
03

Simplify the expression

Next, we simplify the expression \((x^{5})^{5}\). This is done by multiplying the exponents, which gives \(x^{25}\).

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.

Simplifying exponents
Exponentiation involves expressions with exponents, which represent repeated multiplication. Simplifying exponents means expressing these expressions in their simplest form.
When you have multiple expressions with the same base, simplifying involves applying certain rules to compactly express the term.
  • For example, in the expression \( x^2 x^3 \), both terms have the base \(x\).
  • This allows us to use the property of exponents that states when multiplying like bases, you add the exponents.
Thus, \( x^2 x^3 = x^{2+3} = x^5 \). This step reduces the complexity, making further operations much easier. Simplification ensures any expression becomes more refined and straightforward for subsequent operations.
Multiplication of exponents
The multiplication of exponents occurs when you have expressions with the same base. This lets you combine these expressions into a single unit. The key rule to remember is:
  • If you multiply terms with the same base, add the exponents together.
Consider an expression like \( x^a \times x^b \). Here, you can simplify it as \( x^{a+b} \).
  • In the original problem, we encounter \( x^2 x^3 \), both having the base \(x\).
  • According to the multiplication of exponents rule, this gives us \( x^{2+3} = x^5 \).
Multiplication of exponents effectively shortens expressions, ensuring quicker and easier further computations.
Power of a power rule
The power of a power rule is an essential principle of exponentiation. It states that when an exponentiated term is raised to another power, the exponents must be multiplied. This process converts a complicated-looking expression into a more comprehensible form.
  • The rule can be illustrated as: If you have \((x^m)^n\), it simplifies to \(x^{m \times n}\).
In the problem \((x^5)^5\), you apply this rule:
  • Multiply the exponents: \(5 \times 5 = 25\).
  • Hence, \((x^5)^5\) simplifies to \(x^{25}\).
Understanding and applying the power of a power rule helps you manage expressions efficiently, leading to compact and simpler expressions.

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

The revenue \(r\) (in dollars) that a manufacturer of desk chairs receives is given by the polynomial function $$r=f(d)=-0.08 d^{2}+100 d$$ where \(d\) is the number of chairs manufactured. Find the revenue received when 815 chairs are manufactured.

Let \(P(x)=3 x-5 .\) Find each value. $$\text { If } P(x)=x^{23}+5 x^{2}+73 \text { and } Q(x)=x^{23}+4 x^{2}+73, \text { find } P(7)-Q(7)$$

Give an example of a polynomial that is \(\ldots\) of degree 0

Give the degree of each polynomial. $$x^{12}+3 x^{2} y^{3} z^{4}$$

The radius of one pulley in the illustration is 1 inch greater than the radius of the second pulley, and their areas differ by \(4 \pi\) square inches. Find the radius of the smaller pulley.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.