Problem 37 Which is greater? \(\left(2^{3}\... [FREE SOLUTION]

91影视

Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 10: Problem 37

Which is greater? $\left(2^{3}\right)^{2}$ or $\left(2^{2}\right)^{3}$

Short Answer

Expert verified

Both expressions are equal; $ \left( 2^{3} \right)^{2} = \left( 2^{2} \right)^{3} = 2^{6} $.

Step by step solution

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

First, simplify the expression $ \left( 2^{3} \right)^{2} $. Use the property of exponents that $ \left( a^{m} \right)^{n} = a^{m \cdot n} $. Therefore, $ \left( 2^{3} \right)^{2} = 2^{3 \cdot 2} = 2^{6} $.

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

Next, simplify the expression $ \left( 2^{2} \right)^{3} $. Similarly, use the exponent rule that $ \left( a^{m} \right)^{n} = a^{m \cdot n} $. Therefore, $ \left( 2^{2} \right)^{3} = 2^{2 \cdot 3} = 2^{6} $.

Compare the Results

Now compare the simplified expressions. Both $ 2^{6} $ and $ 2^{6} $ are equal, which means $ \left( 2^{3} \right)^{2} $ is equal to $ \left( 2^{2} \right)^{3} $.

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

Simplifying exponents is about making expressions with exponents easier to understand and work with.
For example, let's consider the expression $\left(2^{3}\right)^{2}$.
We use the property of exponents which states that $\left(a^{m}\right)^{n} = a^{m \cdot n}$.
This means we multiply the exponents together: $2^{3 \cdot 2} = 2^{6}$.
Following the same method, we simplify $\left(2^{2}\right)^{3}$ to get $2^{6}$ as well.
Simplifying exponents helps to streamline calculations and make comparisons more straightforward.

properties of exponents

Properties of exponents explain how to manipulate exponential expressions during calculations.
Some key properties include:

\texponentiation to a power: $\left(a^{m}\right)^{n} = a^{m \cdot n}$
\tmultiplication: $a^{m} \cdot a^{n} = a^{m+n}$
\tdivision: $\frac{a^{m}}{a^{n}} = a^{m-n}$
\tzero exponent: $a^{0} = 1$

By understanding these properties, we can more easily simplify and compare exponential expressions, such as we did with $\left(2^{3}\right)^{2}$ and $\left(2^{2}\right)^{3}$.
Each property serves as a useful tool in solving various mathematical problems involving exponents.

comparison of powers

Comparing powers involves evaluating the value of different exponential expressions to see which is greater.
In our exercise, we compared $\left(2^{3}\right)^{2}$ and $\left(2^{2}\right)^{3}$.

Lorem Ipsum. By simplifying them, we found that both are equal, $2^{6} = 2^{6}$. To compare powers effectively:

First, simplify the expressions using exponent properties.
Then, write them in a common form or base.
Finally, compare the results directly.

This method can be used for more complex expressions too, making it a useful skill in prealgebra and beyond.

prealgebra concepts

Prealgebra introduces foundational math skills including exponents, which are crucial for higher-level math.
It covers the basics of working with exponents, such as:
-Understanding what an exponent represents (
e.g., $2^{3} $ means 2 multiplied by itself 3 times: $2 \cdot 2 \cdot 2$).
-Simplifying expressions using exponent rules.
-Learning the properties of exponents to manipulate and compare terms.

Grasping these prealgebra concepts is essential for students to progress in mathematics, as they form the basis of more complex topics in algebra and calculus.

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

Which is greater? \(\left(2^{3}\right)^{2}\) or \(\left(2^{2}\right)^{3}\)

Short Answer

Step by step solution

Simplify \( \left(2^{3}\right)^{2} \)

Simplify \( \left( 2^{2} \right)^{3} \)

Compare the Results

Key Concepts

simplifying exponents

properties of exponents

comparison of powers

prealgebra concepts

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.