/*! 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 44 Use scientific notation and the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 44

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{(0.00016)(300)(0.028)}{0.064}\)

Short Answer

Expert verified
The result is \(2.1 \times 10^{-2}\).

Step by step solution

01

Convert to Scientific Notation

Express each of the numbers in the expression using scientific notation. We have:\(0.00016 = 1.6 \times 10^{-4}\), \(300 = 3 \times 10^2\), \(0.028 = 2.8 \times 10^{-2}\), and \(0.064 = 6.4 \times 10^{-2}\).
02

Set Up the Fraction

Substitute these values into the original fraction, yielding:\[\frac{(1.6 \times 10^{-4})(3 \times 10^2)(2.8 \times 10^{-2})}{6.4 \times 10^{-2}}\].
03

Multiply the Numerator

Calculate the product of the coefficients and the powers of ten separately in the numerator:\(1.6 \times 3 \times 2.8 = 13.44\)and \(10^{-4} \times 10^{2} \times 10^{-2} = 10^{-4 + 2 - 2} = 10^{-4}\).So, the numerator becomes \(13.44 \times 10^{-4}\).
04

Simplify the Expression

Now divide the product by the denominator:\[\frac{13.44 \times 10^{-4}}{6.4 \times 10^{-2}}\].Separate the coefficients and the powers of ten:\(\frac{13.44}{6.4} = 2.1\) and \(\frac{10^{-4}}{10^{-2}} = 10^{-4 + 2} = 10^{-2}\).Thus, the fraction simplifies to \(2.1 \times 10^{-2}\).

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
In mathematics, exponents are used to express repeated multiplication of a number by itself. Understanding the properties of exponents can simplify complex problems, especially in scientific notation. Here are some basic properties:
  • Product of Powers: Add the exponents when multiplying two bases of the same base, i.e., \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: Subtract the exponents when dividing two bases of the same base, i.e.,\(a^m / a^n = a^{m-n}\).
  • Power of a Power: Multiply the exponents when raising a power to another power, i.e.,\((a^m)^n = a^{m \times n}\).
Learn these properties thoroughly, as they serve as foundational tools for working with scientific notation.
Multiplication of Scientific Notation
When multiplying numbers in scientific notation, break it down into manageable steps. Use the properties of exponents to guide you:
  • Multiply the coefficients: If given two numbers in scientific notation \( (a \times 10^m) \) and \( (b \times 10^n) \), first calculate \( a \times b \).
  • Add the exponents: Next, handle the \( 10^m \times 10^n \) part by adding exponents, e.g., \( 10^{m+n} \).
Combining these steps simplifies expression multiplication considerably. For example, \((1.6 \times 10^{-4})(3 \times 10^2) = 4.8 \times 10^{-2}\). This simplification makes handling very large or small numbers easier.
Division of Scientific Notation
Dividing numbers in scientific notation requires a similar process to multiplication, but with division rules:
  • Divide the coefficients: With numbers \( (a \times 10^m) \) and \( (b \times 10^n) \), compute \( a / b \).
  • Subtract the exponents: For \( 10^m / 10^n \), subtract the exponents, i.e., \( 10^{m-n} \).
This allows for maintaining a precise form for complex numbers. For instance, \( \frac{13.44 \times 10^{-4}}{6.4 \times 10^{-2}} = 2.1 \times 10^{-2} \). Following these steps makes operations on scientific notation a breeze, ensuring clarity in your calculations.
Simplifying Scientific Notation Expressions
Simplifying expressions in scientific notation involves using the principles of multiplication and division. Begin by handling the coefficients and exponents separately for clarity.
  • Organize your work: Start by writing out the full expression in scientific notation.
  • Perform operations: Multiply or divide the coefficients, then add or subtract exponents as guided by the operation.
  • Combine and simplify: Reassemble the expression, ensuring the coefficient is within the typical scientific notation range (between 1 and 10).
Once simplified, expressions like \(2.1 \times 10^{-2}\) are more interpretable. Proper simplification makes calculations more efficient and results clearer.

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

Sometimes it is more convenient to express a number as a product of a power of 10 and a number that is not between 1 and 10 . For example, suppose that we want to calculate \(\sqrt{640,000}\). We can proceed as follows: $$ \begin{aligned} \sqrt{640,000} &=\sqrt{(64)(10)^{4}} \\ &=\left((64)(10)^{4}\right)^{\frac{1}{2}} \\ &=(64)^{\frac{1}{2}}\left(10^{4}\right)^{\frac{1}{2}} \\ &=(8)(10)^{2} \\ &=8(100)=800 \end{aligned} $$ Compute each of the following without a calculator, and then use a calculator to check your answers. (a) \(\sqrt{49,000,000}\) (b) \(\sqrt{0.0025}\) (c) \(\sqrt{14,400}\) (d) \(\sqrt{0.000121}\) (e) \(\sqrt[3]{27,000}\) (f) \(\sqrt[3]{0.000064}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{3}}{\sqrt[4]{3}}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{66,000,000,000}{0.022}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(9 x^{2} y^{4}\right)^{\frac{1}{2}}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \((0.000004)(120,000)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.