/*! 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 81 Show that \(\left(3^{\sqrt{2}}\r... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 81

Show that \(\left(3^{\sqrt{2}}\right)^{\sqrt{2}}=9\).

Short Answer

Expert verified
We are given the expression \(\left(3^{\sqrt{2}}\right)^{\sqrt{2}}\). Using the property of exponents \((a^b)^c = a^{bc}\), we get \(3^{\sqrt{2} \cdot \sqrt{2}}\). Simplifying the exponent, we find \(\sqrt{2} \cdot \sqrt{2} = 2\), so our expression becomes \(3^2\). Calculating the value of \(3^2\), we find it is equal to 9. Thus, we have shown that \(\left(3^{\sqrt{2}}\right)^{\sqrt{2}}=9\).

Step by step solution

01

Identify the given expression

We are given the expression \(\left(3^{\sqrt{2}}\right)^{\sqrt{2}}\). Our goal is to simplify it and show that it is equal to 9.
02

Apply the property of exponents

We will use the property of exponents that states \((a^b)^c = a^{bc}\). Applying this rule to our given expression, we get: \[\left(3^{\sqrt{2}}\right)^{\sqrt{2}} = 3^{\sqrt{2} \cdot \sqrt{2}}\]
03

Simplify the exponent

Now, we will simplify the exponent by multiplying the two square roots: \[\sqrt{2} \cdot \sqrt{2} = (\sqrt{2})^2 = 2\] So, our expression becomes: \[3^{\sqrt{2} \cdot \sqrt{2}} = 3^2\]
04

Calculate the result

Finally, we calculate the value of \(3^2\): \[3^2 = 9\] Thus, we have shown that \(\left(3^{\sqrt{2}}\right)^{\sqrt{2}}=9\).

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.

Properties of Exponents
Exponents, sometimes called powers or indices, are a shorthand way of expressing repeated multiplication of a number by itself. Understanding the properties of exponents is key to simplifying expressions easily. These properties allow us to manipulate exponential expressions in various ways, making equations easier to solve or simplify. Here are some primary properties:
  • Product of Powers Property: This states that when you multiply like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Power of a Power Property: This involves raising an expression with an exponent to another power. The rule is \((a^m)^n = a^{mn}\).
  • Power of a Product Property: When distributing an exponent over a product, you apply the exponent to each factor in the product: \((ab)^m = a^m b^m\).
In our original exercise, we specifically used the 'Power of a Power Property' to effectively combine the exponents.
Simplifying Expressions
Simplifying expressions is an important skill in mathematics. It involves reducing an expression to its simplest form while maintaining its original value. The goal is to make calculations easier and ensure equations are more manageable.When simplifying expressions involving exponents, you can use the properties of exponents to combine and reduce terms:
  • First, apply known properties of exponents to rewrite the expression.
  • Identify common bases to combine terms when necessary.
  • Calculate any numerical results such as integer powers or square roots.
For instance, in our exercise, we started with \(\left(3^{\sqrt{2}}\right)^{\sqrt{2}}\) and cleverly simplified it using multiplication of the exponents.
Power of a Power Rule
This essential rule of exponents simplifies expressions where you have a power raised to another power. The 'Power of a Power Rule' can be expressed as:\[(a^m)^n = a^{m \cdot n}\]This means when you have an exponent raised to another exponent, you multiply the exponents. This rule is particularly useful for simplifying expressions without needing to expand them fully.In the given exercise, we applied the rule to \((3^{\sqrt{2}})^{\sqrt{2}}\).
  • The expression was initially in the form of a power raised to another power.
  • We multiplied the two exponents together, \(\sqrt{2} \cdot \sqrt{2} = 2\), making the expression \(3^2\).
  • We further calculated this to get the simplified result of 9.
By using the 'Power of a Power Rule', we transformed a complex-looking expression into a simple integer, illustrating the power of this tool 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 a description of how the shape of the St. Louis Gateway Arch is related to the graph of \(\cosh x\). You should be able to find the necessary information using an appropriate web search.

For \(x=3\) and \(y=8\), evaluate each of the following: (a) \(\ln (x y)\) (b) \((\ln x)(\ln y)\)

Find the number \(t\) that makes \(e^{t^{2}+6 t}\) as small as possible. $$ \text { [Here } e^{t^{2}+6 t} \text { means } e^{\left(t^{2}+6 t\right)} \text { .] } $$

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=5 e^{9 x}\)

Estimate the indicated value without using a calculator. \(e^{-0.00046}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.