Problem 138 Evaluate. See Sections 1.3 and 5... [FREE SOLUTION]

Chapter 5: Problem 138

Evaluate. See Sections 1.3 and 5.1 $$ \left(2^{2}\right)^{3} $$

Short Answer

Expert verified

The value of $ (2^2)^3 $ is 64.

Step by step solution

Understand the Expression

The given expression is $ \left(2^2\right)^3 $. This involves the operation of exponents, specifically a power of a power.

Apply the Power of a Power Rule

According to the power of a power rule for exponents, $ (a^m)^n = a^{m \cdot n} $. In this case, $ a = 2 $, $ m = 2 $, and $ n = 3 $. We apply the rule: $ (2^2)^3 = 2^{2 \cdot 3} $.

Simplify the Exponent

Calculate the new exponent: $ 2 \times 3 = 6 $. Thus, $ 2^{2 \cdot 3} = 2^6 $.

Calculate the Final Result

Now, calculate $ 2^6 $. We multiply 2 by itself 6 times: $ 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 $. Hence, $ 2^6 = 64 $.

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.

Power of a Power Rule

When dealing with exponents, it's important to understand the 'power of a power rule'. This rule is used when you have one exponent raised to another exponent, like this \[ (a^m)^n \]. According to the rule, you simplify this by multiplying the exponents:

Keep the base the same (a)
Multiply the exponents (m) and (n) together
Your final expression becomes $ a^{m \cdot n} $

For example, let's take $(2^2)^3$. Here, the base "a" is 2, "m" is 2, and "n" is 3. When you apply the rule, $(2^2)^3 = 2^{2 \cdot 3} = 2^6$. This step greatly simplifies finding the final value of the expression.

Simplifying Exponents

Once you have adjusted your exponents using the power of a power rule, the next step is simplifying. Sometimes, especially with larger bases or exponents, finding the product of your exponents can look complicated, but stay calm!

Always start by performing the multiplication of the exponents. In our example, $2 \cdot 3 = 6$.
Next, simplify the base raised to the new power. For $2^6$, it's a matter of multiplying 2 by itself until you鈥檝e done it 6 times.
To make the process easy under pressure, break down steps: calculate $2^2$ first, then continue.鈥�

For $2^6$, you sequentially calculate it as follows: $2 \times 2 = 4$, then $4 \times 2 = 8$, and so on, until you reach the sixth multiplication. Simplifying exponents often ends up quicker than thought once you block this process in parts.

Evaluating Expressions

After understanding and applying rules for simplifying exponents, evaluating the expression becomes the final task. Why is this important? Because it provides you with a tangible result.

The final number gives us an answer to the question at hand: "What does this exponential statement equal to?"
The steps of evaluation involve computing the final result after simplification, which requires basic multiplication of integers.

In our example of $2^6$, evaluation involves computing $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$. This step not only finalizes your work but reassures understanding of both the mathematical concept and its application. Each segment of solving eventually assists in mastering exponential expressions, thereby preparing you for more complex problems ahead.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate. See Sections 1.3 and 5.1 $$ \left(2^{2}\right)^{3} $$

Short Answer

Step by step solution

Understand the Expression

Apply the Power of a Power Rule

Simplify the Exponent

Calculate the Final Result

Key Concepts

Power of a Power Rule

Simplifying Exponents

Evaluating Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Applied Mathematics

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.