/*! 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 23 Find integers \(m\) and \(n\) su... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 23

Find integers \(m\) and \(n\) such that \(2^{m} \cdot 5^{n}=16000\).

Short Answer

Expert verified
The integers $m$ and $n$ satisfying the given equation \(2^m \cdot 5^n=16000\) are \(m = 7\) and \(n = 3\).

Step by step solution

01

Prime factorize 16000

To find the correct exponents for 2 and 5, we need to prime factorize the number 16000. When prime factorized, the number will be represented as a product of powers of prime factors. \( 16000 = 2^4 \cdot 1000 = 2^4 \cdot 2^3 \cdot 125 = 2^7 \cdot 5^3 \)
02

Match the prime factors to the given expression

Now that we have the prime factors, we can compare them to the given expression \(2^m \cdot 5^n\). We can see that in the factorization, \(2^7\) represents the first part of the expression, and \(5^3\) represents the second part.
03

Identify the values of m and n

Based on the prime factorization we obtained, we have \(2^7 \cdot 5^3 \). So we can deduce the values of m and n as follows: \( m = 7 \) \( n = 3 \) So the integers m and n satisfying the given equation \(2^m \cdot 5^n=16000\) are \(m = 7\) and \(n = 3\).

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!

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

Suppose \(t(x)=\frac{5}{4 x^{3}+3}\). (a) Show that the point (-1,-5) is on the graph of \(t\) (b) Give an estimate for the slope of a line containing (-1,-5) and a point on the graph of \(t\) very close to (-1,-5)

Find a polynomial \(p\) of degree 3 such that \(-1,2,\) and 3 are zeros of \(p\) and \(p(0)=1\).

Suppose \(r\) is the function with domain \((0, \infty)\) defined by $$ r(x)=\frac{1}{x^{4}+2 x^{3}+3 x^{2}} $$ for each positive number \(x\). (a) Find two distinct points on the graph of \(r\). (b) Explain why \(r\) is a decreasing function on \((0, \infty)\). (c) Find two distinct points on the graph of \(r^{-1}\).

Explain why the polynomial \(p\) defined by $$ p(x)=x^{6}+7 x^{5}-2 x-3 $$ has a zero in the interval (0,1) .

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ \frac{s(1+x)-s(1)}{x} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.