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Chapter 1: Problem 44

Approximate the real-number expression. Express the answer in scientific notation accurate to four significant figures. A. \(\sqrt{\left|3.45-1.2 \times 10^{4}\right|+10^{5}}\) B. \(\left(1.79 \times 10^{2}\right) \times\left(9.84 \times 10^{3}\right)\)

Short Answer

Expert verified
A: \(3.346 \times 10^2\), B: \(1.762 \times 10^6\).

Step by step solution

01

Evaluate Inside the Absolute Value

For expression A, we first evaluate the term inside the absolute value, \(3.45 - 1.2 \times 10^4\). Calculating this, we have \(-11,996.55\). The absolute value of \(-11,996.55\) is \(|-11,996.55| = 11,996.55\).
02

Sum with Additional Term

Now, add \(11,996.55\) from the absolute value to \(10^5\). This gives us \(11,996.55 + 100,000 = 111,996.55\).
03

Calculate the Square Root

Take the square root of \(111,996.55\). Using a calculator, this results in \( ext{about } 334.60\).
04

Convert to Scientific Notation

Convert \(334.60\) to scientific notation with four significant figures. This is \(3.346 \times 10^2\).
05

Multiply the Numbers

For expression B, multiply \(1.79 imes 10^2\) and \(9.84 imes 10^3\). Apply the property of multiplication in scientific notation: multiply the coefficients \(1.79 imes 9.84\) which is \( ext{about } 17.6236 \) and add the exponents \(2 + 3 = 5\).
06

Convert Product to Scientific Notation

Convert \(17.6236 \times 10^5\) to scientific notation with four significant figures. This results in \(1.762 \times 10^6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
The absolute value of a number is a concept in mathematics that represents the distance of a number from zero on a number line, regardless of its direction. It is denoted by two vertical bars, like this: \(|x|\).
For example:
  • The absolute value of -5 is \(|-5| = 5\), simply because -5 is 5 units away from zero.
  • The absolute value of 7 is \(|7| = 7\), since it is already positive.
In the original exercise, we dealt with the expression \(3.45 - 1.2 \times 10^4\), which evaluates to \(-11,996.55\). The absolute value of this negative number is \(11,996.55\).
This step was crucial for correctly adding it to \(10^5\) in the next part of the calculation.
Square Root
The square root operation finds a number which, when multiplied by itself, yields the original number. It is represented by the radical symbol \(\sqrt{}\).
For example:
  • The square root of 25 is 5 because \(5 \times 5 = 25\).
  • Similarly, the square root of 49 is 7.
In the problem, after calculating the absolute value and adding it with \(10^5\), we arrived at a sum of \(111,996.55\). The next step was to find the square root of this term. Using a calculator, this resulted in approximately \(334.60\).
This value was then converted to scientific notation.
Multiplication in Scientific Notation
Scientific notation is a way to express very large or very small numbers conveniently. It is represented in the form \((a \times 10^n)\), where \(1 \leq |a| < 10\).
When multiplying numbers in scientific notation, the process is straightforward:
  • Multiply the decimal numbers (known as the coefficients).
  • Add the exponents of the powers of ten.
In part B of the original exercise, we multiplied \(1.79 \times 10^2\) with \(9.84 \times 10^3\):
  • The coefficients \(1.79\) and \(9.84\) were multiplied to get \(17.6236\).
  • The exponents \(2\) and \(3\) were added to make \(10^5\).
This yielded the initial result \(17.6236 \times 10^5\), which was then adjusted to proper scientific notation.
Significant Figures
Significant figures in a number reflect precision. These include all the non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion.
Why are significant figures important?
  • They communicate the precision of measurements and calculations.
  • In scientific calculations, they help maintain and convey accuracy.
In the context of this exercise:
  • Expression A resulted in \(334.60\) after finding the square root, which was then expressed as \(3.346 \times 10^2\) to four significant figures.
  • Expression B required the final result \(17.6236 \times 10^5\) to be simplified to \(1.762 \times 10^6\), ensuring the solution maintains four significant figures.
Correctly applying significant figures ensures that the computed results are as precise as possible in scientific terms.

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