/*! 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 14 Evaluate each exponential expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 14

Evaluate each exponential expression in Exercises 1鈥�22. $$3^{3} \cdot 3^{2}$$

Short Answer

Expert verified
The result of the expression \(3^{3} \cdot 3^{2}\) is 243.

Step by step solution

01

Understanding the problem

We are given an exponential expression \(3^{3} \cdot 3^{2}\). This operation involves the same base (3) raised to different powers, 3 and 2.
02

Applying the law of exponents

The law of exponents states that when multiplying like bases, the exponents should be added. This can be expressed as \(a^{m} \cdot a^{n} = a^{m+n}\). Therefore, our expression changes to \(3^{3+2}\).
03

Calculating the result

We now perform the addition of the exponents, leading to \(3^{5}\). This means we have 3 multiplied by itself 5 times. Therefore, the result of our expression is \(3^{5} = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243\).

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.

Exponential Expression
An exponential expression is a mathematical notation that involves numbers known as bases raised to a certain power called exponents. In this expression, the base is the number that is multiplied by itself, while the exponent indicates how many times the base is used as a factor.
In our original example, the exponential expression is given as \(3^3 \cdot 3^2\). Here, 3 is the base and it is raised to the powers of 3 and 2.
This means the base (3) will be multiplied itself according to the respective exponents:
  • \(3^3\) means \(3 \cdot 3 \cdot 3\).
  • \(3^2\) means \(3 \cdot 3\).
Such expressions allow us to simplify the representation of repeated multiplication of the same number, making it easier to handle in calculations and algebraic expressions.
Laws of Exponents
The laws of exponents are essential rules in algebra that help us simplify expressions and solve equations involving powers. These rules dictate how we handle operations of exponentiation efficiently. Let's focus on one of these laws that applies to our problem.
The multiplication of exponents law states:
  • When you multiply two exponential expressions with the same base, you keep the base and add the exponents: \(a^m \cdot a^n = a^{m+n}\).
This simplifies calculational work by reducing a potentially complex problem into manageable steps.
To exemplify this, in our original problem, we multiply two exponents with the base 3. Applying the law, we add the exponents (3 and 2) instead of performing each multiplication separately, simplifying the expression to \(3^{3+2}\).
Multiplication of Exponents
Multiplication of exponents occurs when dealing with expressions sharing a common base. Knowing how to correctly combine these is crucial for simplifying expressions and calculating results swiftly.
For an expression like \(3^3 \cdot 3^2\):
  • Recognize that both numbers have the same base, 3.
  • Instead of multiplying \(3 \cdot 3 \cdot 3\) and \(3 \cdot 3\) separately, we apply the law of exponents to keep computations simple.
We combine the exponents to get \(3^{3+2} = 3^5\), meaning 3 is multiplied by itself five times.
This allows for a quick calculation yielding \(3^5 = 243\). Multiplying exponents in this way not only reduces errors but also saves time during calculations.

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

In Exercises 132鈥�135, determine whether each statement makes sense or does not make sense, and explain your reasoning. There are many exponential expressions that are equal to \(36 x^{12},\) such as \(\left(6 x^{6}\right)^{2},\left(6 x^{3}\right)\left(6 x^{9}\right), 36\left(x^{3}\right)^{9},\) and \(6^{2}\left(x^{2}\right)^{6}\)

Simplify by reducing the index of the radical. $$\sqrt[9]{x^{6} y^{3}}$$

Explain how to factor \(3 x^{2}+10 x+8\)

Factor completely. $$ -x^{2}-4 x+5 $$

Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. $$\text { a. } 3^{\frac{1}{2}} \square 3^{\frac{1}{3}}$$ $$\text { b. } \sqrt{7}+\sqrt{18} \square \sqrt{7}+18$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.