/*! 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 529 Simplify each expression. $$\l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 529

Simplify each expression. $$\left(r^{3}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \(r^{6}\).

Step by step solution

01

Identify the power of a power rule

The expression \(\big(r^{3}\big)^{2}\) involves an exponent raised to another exponent. Recognize that you can use the power of a power rule, which states \( (a^m)^n = a^{m \times n} \).
02

Apply the power of a power rule

Apply the power of a power rule to simplify \(\big(r^{3}\big)^{2}\). This gives: \((r^{3})^{2} = r^{3 \times 2} \).
03

Multiply the exponents

Multiply the exponents \(3 \times 2\), which results in \(r^{6}\).
04

Write the final simplified expression

Write the final simplified expression as \(r^{6}\).

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.

Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, instead of writing 5 * 5 * 5, we can write 5^3. The small number to the top right of a base number is called an 'exponent,' and it tells us how many times to multiply the base by itself. In our example, 5 is the base, and 3 is the exponent. This means: 5^3 = 5 * 5 * 5 = 125. Exponents make it much easier to write and calculate large numbers.
The expression \((r^{3})^{2}\) etates this way of thinking. You are asked to apply exponentiation twice. This can often be confusing, but we've covered the rules to simplify it.
Simplifying Expressions with Exponents
Simplifying expressions that include exponents is straightforward when following mathematical rules. The 'power of a power' rule is especially helpful. This rule states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
In our exercise, we start with \((r^3)^2\). Here, the base is 'r,' the first exponent is 3, and the second exponent is 2. According to the 'power of a power' rule: \((r^3)^2 = r^{3 \times 2} ;= r^6\).
By following this formula, you simplified the double exponentiation into a single exponentiation.
Applying Mathematical Rules
Mathematical rules are there to make our calculations straightforward and reliable. Here are some key rules that we're using in this exercise:
  • The 'Power of a Power' Rule: \((a^m)^n = a^{m \times n}\)
  • The 'Multiplication' Rule for Exponents: When multiplying two expressions with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\)
  • The 'Division' Rule for Exponents: When dividing two expressions with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\)

Understanding these rules helps you easily break down and simplify complex mathematical expressions. Applying the right rule at the right time can make a huge difference in solving challenges more effectively.

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

Write each number in scientific notation. According to www.cleanair.com, U.S. businesses use about 21,000,000 tons of paper per year.

Simplify each expression. $$\left(5 a^{2}+2 a-12\right)+\left(9 a^{2}+8 a-4\right)$$

A hiker drops a pebble from a bridge 240 feet above a canyon. The polynomial \(-16 t^{2}+240\) gives the height of the pebble \(t\) seconds a after it was dropped. Find the height when \(t=3\) .

Simplify. $$\frac{\left(a^{3}\right)^{2}}{a^{4}}$$

Divide the monomials. $$\frac{\left(5 m^{9} n^{3}\right)\left(8 m^{3} n^{2}\right)}{\left(10 m n^{4}\right)\left(m^{2} n^{5}\right)}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.