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Chapter 1: Problem 44

Approximate the real-number expression. Express the answer in sclentific notation accurate to four significant figures. (a) \(\sqrt{\left|3.45-1.2 \times 10^{4}\right|+10^{3}}\) (b) \(\left(1.791 \times 10^{2}\right) \times\left(9.84 \times 10^{3}\right)\)

Short Answer

Expert verified
(a) \(1.140 \times 10^{2}\); (b) \(1.761 \times 10^{6}\).

Step by step solution

01

Calculate Absolute Value and Sum

First, we need to evaluate the expression inside the square root.Calculate the absolute value: \[\left|3.45 - 1.2 \times 10^{4}\right| = \left|3.45 - 12000\right| = \left|-11996.55\right| = 11996.55\]Add 1000 to the absolute value:\[11996.55 + 1000 = 12996.55\]
02

Compute the Square Root

With the result from Step 1, we now compute the square root:\[\sqrt{12996.55} \approx 114.0066\]Express this in scientific notation to four significant figures:\[1.140 \times 10^{2}\]
03

Multiply the Scientific Notations

Now, we calculate part (b) by multiplying the two scientific notations:First, calculate the coefficients:\[1.791 \times 9.84 = 17.61264\]Then multiply the powers of ten:\[10^{2} \times 10^{3} = 10^{5}\]Combine these results:\[17.61264 \times 10^{5}\]
04

Adjust to Scientific Notation

Finally, express the result from Step 3 in scientific notation with four significant figures:\[1.761 \times 10^{6}\]

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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 like finding the distance from zero on the number line. This concept is crucial because it helps us measure how large a number is, regardless of its sign. Whether positive or negative, the absolute value converts it to a positive number.

For instance, in the exercise given, we calculate the absolute value of the expression \(3.45 - 1.2 \times 10^{4}\). Breaking it down, first calculate \(3.45 - 12000\), which equals \(-11996.55\). The absolute value of this result, \|-11996.55\|, is simply \(11996.55\). The purpose of taking the absolute value is to ignore the negative sign, simply focusing on the magnitude.

Key points about absolute value:
  • Represents only magnitude, not direction.
  • Always non-negative.
  • Useful for distance calculations.
Square Root
A square root is a special value that, when multiplied by itself, gives the original number. Understanding square roots is important in many areas of math, including geometry and algebra.

In the exercise, after evaluating the absolute value and adding 1000, we take the square root of \(12996.55\). This step is crucial as it transforms the sum into a value that we can further use in scientific notation. The approximate value of \(\sqrt{12996.55}\) is \(114.0066\).

Important aspects of square roots:
  • Only positive numbers have real square roots.
  • Indicated by the radical symbol \(\sqrt{}\).
  • Square roots of perfect squares can be integers.
Significant Figures
Significant figures are the digits in a number that contribute to its precision. In scientific calculations, we often use significant figures to ensure our results have the proper level of accuracy.

In the problem, it's important to express results to four significant figures. For example, the square root result of \(114.0066\) is expressed as \(1.140 \times 10^{2}\), maintaining four significant figures.

How to identify significant figures:
  • All non-zero digits are significant.
  • Any zeros between significant figures are also significant.
  • Leading zeros are not significant.
  • Trailing zeros in a decimal number are significant.
Multiplying Powers of Ten
Multiplying powers of ten is a fundamental step in handling scientific notation. Scientific notation helps us to express very large or very small numbers more conveniently.

In the exercise, we multiply \(1.791 \times 10^{2}\) by \(9.84 \times 10^{3}\). First, you multiply the coefficients \(1.791 \times 9.84\) to get \(17.61264\). Then, for the powers of ten, you add the exponents: \(10^{2} \times 10^{3} = 10^{5}\). The final result before adjusting for significant figures is \(17.61264 \times 10^{5}\).

Steps to multiply powers of ten:
  • Multiply the base numbers (coefficients) first.
  • Add the exponents when multiplying the powers of ten.
  • Express the result in proper scientific notation ensuring correct significant figures.

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Most popular questions from this chapter

A person's body surface area \(S\) (in square feet) can be approximated by $$S=(0.1091) w^{1423} h^{0.723}$$ where height \(h\) is in inches and weight \(w\) is in pounds. (a) Estimate \(S\) for a person 6 feet tall weighing 175 pounds. (b) If a person is 5 feet 6 inches tall, what effect does a \(10 \%\) increase in weight have on \(S ?\)

The table contains average annual temperatures for the northern and southern hemispheres at various latitudes. $$\begin{array}{|c|c|c|}\hline \text { Latitude } & \text { N. hem. } & \text { S. hem. } \\\\\hline 85^{\circ} & -8^{\circ} \mathrm{F} & -5^{\circ} \mathrm{F} \\\\\hline 75^{\circ} & 13^{\circ} \mathrm{F} & 10^{\circ} \mathrm{F} \\\\\hline 65^{\circ} & 30^{\circ} \mathrm{F} & 27^{\circ} \mathrm{F} \\\\\hline 55^{\circ} & 41^{\circ} \mathrm{F} & 42^{\circ} \mathrm{F} \\\\\hline 45^{\circ} & 57^{\circ} \mathrm{F} & 53^{\circ} \mathrm{F} \\\\\hline 35^{\circ} & 68^{\circ} \mathrm{F} & 65^{\circ} \mathrm{F} \\\\\hline 25^{\circ} & 78^{\circ} \mathrm{F} & 73^{\circ} \mathrm{F} \\\\\hline 15^{\circ} & 80^{\circ}\mathrm{F} & 78^{\circ} \mathrm{F} \\\\\hline 5^{\circ} & 79^{\circ} \mathrm{F} & 79^{\circ} \mathrm{F} \\\\\hline\end{array}$$ (a) Which of the following equations more accurately predicts the average annual temperature in the southern hemisphere at latitude \(L ?\) (1) \(T_{1}=-1.09 L+96.01\) (2) \(T_{2}=-0.011 L^{2}-0.126 L+81.45\) (b) Approximate the average annual temperature in the southern hemisphere at latitude \(50^{\circ} .\)

Choose the equation that best describes the table of data. (Hint: Make assignments to \(\mathbf{Y}_{\mathbf{r}}-\mathbf{Y}_{\mathbf{4}}\) and examine a table of their values.) $$\begin{array}{|c|c|}\hline x & y \\\\\hline 1 & -9 \\\\\hline 2 & -4 \\\\\hline 3 & 11 \\\\\hline 4 & 42 \\\\\hline 5 & 95 \\\\\hline\end{array}$$ (1) \(y=13 x-22\) (2) \(y=x^{2}-2 x-8\) (3) \(y=4 \sqrt{x}-13\) (4) \(y=x^{3}-x^{2}+x-10\)

The average weight \(W\) (in pounds) for men with height \(h\) between 64 and 79 inches can be approximated using the formula \(W=0.1166 h^{17}\). Construct a table for \(W\) by letting \(h=64,65, \ldots, 79 .\) Round all weights to the nearest pound. (TABLE CANT COPY)

Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$\left(a^{2}+1\right)^{1 / 2} \square a+1$$
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