/*! 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 590 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 590

In the following exercises, simplify. $$ 4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} $$

Short Answer

Expert verified
The simplified form is 64.

Step by step solution

01

Apply the Product of Powers Rule

The product of powers rule states that when multiplying two exponents with the same base, you add the exponents. The base here is 4. So, we have:\[4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} = 4^{\left(\frac{5}{2} + \frac{1}{2}\right)}\]
02

Add the Exponents

Add the exponents \(\frac{5}{2}\) and \(\frac{1}{2}\):\[\frac{5}{2} + \frac{1}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3\]
03

Simplify the Expression

Now substitute the result of the addition back into the exponent, giving:\[4^3\]
04

Compute the Final Value

Calculate \(4^3\):\[4^3 = 4 \cdot 4 \cdot 4 = 64\]

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.

Product of Powers Rule
The product of powers rule is a fundamental concept in exponentiation. It states that when you multiply two exponents with the same base, you simply add their exponents. This works because you are essentially combining the repeated multiplications.
For example:
\(a^m \cdot a^n = a^{m+n}\)
In the exercise, we applied this rule to the base 4:
\[4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} = 4^{(\frac{5}{2} + \frac{1}{2})}\]
This addition of exponents reduces the problem to just dealing with one exponent, simplifying the calculation process significantly.
Simplifying Exponents
Simplifying exponents is a necessary skill to manage complex expressions easily. In this case, after applying the product of powers rule, we simplified the expression further by adding the exponents.
Adding exponents like \(\frac{5}{2}\) and \(\frac{1}{2}\) involves using basic fraction addition rules. We can express these fractions with a common denominator and then add them together:
\[\frac{5}{2} + \frac{1}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3\]
Thus, the exponent becomes 3, transforming \(4^{\frac{6}{2}}\) into \(4^3\). Removing fractions from the exponent makes further calculations straightforward.
Adding Fractions
Adding fractions is a basic yet crucial arithmetic skill. When adding fractions, you add the numerators (the top numbers) while keeping the same denominator (the bottom number).
For instance, in our exercise, we added \(\frac{5}{2}\) and \(\frac{1}{2}\):
\[\frac{5}{2} + \frac{1}{2} = \frac{5+1}{2} = \frac{6}{2}\]
Once we have the sum, simplifying the resulting fraction to its simplest form gives us the next step for our exponentiation. In this case, \(\frac{6}{2}\) simplifies to 3, making the expression much easier to compute.

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 the following exercises, simplify and rationalize the denominator. $$ -\frac{9}{2 \sqrt{3}} $$

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{7}{40}} $$

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula \(s=\sqrt{A}\) to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.

In the following exercises, simplify and rationalize the denominator. $$ -\frac{8}{3 \sqrt{6}} $$

In the following exercises, solve. $$ 2 \sqrt{8 r+1}-8=2 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.