/*! 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 530 Simplify each expression. $$\l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 530

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

Short Answer

Expert verified
3^{10}

Step by step solution

01

Identify the Power Rule

The expression \((3^2)^5\) involves an exponent raised to another exponent. Use the power rule of exponents which states \[ (a^m)^n = a^{m \times n} \] to simplify the expression.
02

Apply the Power Rule

Identify the base (3) and the exponents (2 and 5). Multiply the two exponents together: \[ 2 \times 5 = 10 \]
03

Simplify the Expression

Rewrite the expression using the result from Step 2: \[ 3^{10} \]
04

Final Simplified Form

The simplified form of the expression is \[ 3^{10} \].

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 represent repeated multiplication. For example, instead of writing 3 multiplied by itself 4 times, we can write it as \3^4\. Here, 3 is called the base, while 4 is the exponent.
Exponents follow certain rules that help us simplify expressions. One important rule is the Power Rule, which is used when an exponent is raised to another exponent. This rule states \( (a^m)^n = a^{m \times n} \). This rule is very handy for simplifying nested exponentiated terms.
In short, remember:
  • The base is the number being multiplied.
  • The exponent tells how many times to multiply the base by itself.
  • Exponents make writing and calculating large numbers easier.
Simplifying Expressions
Simplifying expressions involves using rules like the Power Rule to make expressions easier to work with. In our original exercise \( (3^2)^5 \), we used the Power Rule to combine the exponents.
To apply the Power Rule, follow these steps:
  • Identify the base and both exponents. Here, the base is 3, and the exponents are 2 and 5.
  • Multiply the two exponents together: \ 2 \times 5 = 10 \.
  • Rewrite the expression using the new exponent: \ 3^{10} \.
By following these steps, we simplify the expression and make it manageable. Simplifying expressions helps in solving equations and making calculations more straightforward.
Mathematical Operations
Mathematical operations include addition, subtraction, multiplication, and division. Exponentiation is also a key operation, which deals with repeated multiplication.
To handle expressions with exponents, you must understand a few basic rules:
  • Power Rule: As seen \( (a^m)^n = a^{m \times n} \).
  • Product Rule: When multiplying bases with exponents \( a^m \times a^n = a^{m + n} \).
  • Quotient Rule: When dividing bases with exponents \( \frac{a^m}{a^n} = a^{m-n} \).
  • Zero Exponent Rule: Any base to the zero power is 1 \ a^0 = 1 \.
With a good grasp of these rules, you can easily tackle any exponent-based problem. Practice makes perfect, so make sure to apply these rules often!

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

Simplify. $$\left(m^{5}\right)^{-1}$$

Simplify each expression. $$n \cdot n^{4}$$

Simplify. $$\frac{n^{3}}{n^{12}}$$

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

Multiply and write your answer in decimal form. $$\frac{9 \times 10^{-3}}{1 \times 10^{-6}}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.