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Chapter 1: Problem 47

Express the following numbers as decimals: (a) \(1.52 \times 10^{-2}\) (b) \(7.78 \times 10^{-8}\) (c) \(1 \times 10^{-6}\) (d) \(1.6001 \times 10^{3}\).

Short Answer

Expert verified
(a) 0.0152, (b) 0.0000000778, (c) 0.000001, (d) 1600.1

Step by step solution

01

Understand Scientific Notation

Scientific notation expresses numbers as a product of a coefficient (a decimal number) and a power of 10. In this case, the formula used is: \(a \times 10^{n}\),where \(a\) is the coefficient and \(n\) is the exponent.
02

Step 2a: Convert (a) \(1.52 \times 10^{-2}\)

To convert \(1.52 \times 10^{-2}\), recognize that it means moving the decimal point 2 places to the left, as the exponent is negative. The number becomes 0.0152.
03

Step 2b: Convert (b) \(7.78 \times 10^{-8}\)

To convert \(7.78 \times 10^{-8}\), move the decimal point 8 places to the left. This results in the number 0.0000000778.
04

Step 2c: Convert (c) \(1 \times 10^{-6}\)

Here, move the decimal point 6 places to the left according to the negative exponent. Thus, the number becomes 0.000001.
05

Step 2d: Convert (d) \(1.6001 \times 10^{3}\)

In this case, move the decimal point 3 places to the right since the exponent is positive. The number becomes 1600.1.

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Key Concepts

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

Decimal Conversion
Decimal conversion involves shifting the decimal point to transform a scientific notation expression into its standard decimal form. Understanding where to move the decimal is crucial. The exponent of 10 dictates the number of places the decimal point shifts, and its direction. For positive exponents, shift to the right, making the number larger. For negative exponents, shift to the left, making the number smaller. These shifts adjust the decimal to represent the full value in its regular form.
For example, converting a scientific notation like \(1.52 \times 10^{-2}\) requires moving the decimal 2 places to the left, resulting in the standard decimal 0.0152.
Negative Exponents
Negative exponents in scientific notation indicate moving the decimal point to the left, effectively making the number smaller than the coefficient. An exponent like \(-n\) means placing the decimal \(n\) positions to the left.
For instance, in \(7.78 \times 10^{-8}\), the negative exponent \(-8\) guides the conversion to 0.0000000778 by moving the decimal left by eight spots. This method helps express tiny fractions easily and concisely in scientific contexts.
Positive Exponents
Positive exponents signify multiplying a number by powers of ten, thus shifting the decimal point right. This operation enlarges the initial coefficient, presenting a much larger number in decimal form. A positive exponent means move the decimal to the right as many places as the exponent indicates.
For example, \(1.6001 \times 10^{3}\) involves moving the decimal three places to the right. This creates a bigger number鈥�1600.1鈥攊n decimal format, simplifying large numbers that may be cumbersome to write otherwise.
Coefficients in Scientific Notation
Coefficients in scientific notation are the numbers multiplied by powers of ten, denoting the core value of the expression. The coefficient should be a decimal number between 1 and 10. Its value stays unchanged during conversion, only the placement of the decimal point alters.
For example, in the expression \(1 \times 10^{-6}\), the coefficient, 1, remains as is while only the decimal position shifts based on the exponent. Understanding the significance of the coefficient allows for a correct conversion whether the exponents are positive or negative.

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Most popular questions from this chapter

The density of ammonia gas under certain conditions is \(0.625 \mathrm{~g} / \mathrm{L} .\) Calculate its density in \(\mathrm{g} / \mathrm{cm}^{3}\).

A student is given a crucible and asked to prove whether it is made of pure platinum. She first weighs the crucible in air and then weighs it suspended in water (density = \(0.9986 \mathrm{~g} / \mathrm{mL}\) ). The readings are \(860.2 \mathrm{~g}\) and \(820.2 \mathrm{~g}\), respectively. Based on these measurements and given that the density of platinum is \(21.45 \mathrm{~g} / \mathrm{cm}^{3},\) what should her conclusion be? (Hint: An object suspended in a fluid is buoyed up by the mass of the fluid displaced by the object. Neglect the buoyancy of air.)

The highest speed limit in the United States is \(85 \mathrm{mph}\) on an isolated stretch of rural interstate in Texas. What is the speed limit in kilometers per hour \((1 \mathrm{mi}=1609 \mathrm{~m})\) ? Report your answer as a whole number.

The unit "troy ounce" is often used for precious metals such as gold \((\mathrm{Au})\) and platinum \((\mathrm{Pt})(1\) troy ounce \(=\) \(31.103 \mathrm{~g}\) ). (a) A gold coin weighs 2.41 troy ounces. Calculate its mass in grams. (b) Is a troy ounce heavier or lighter than an ounce \((1 \mathrm{lb}=16 \mathrm{oz} ; 1 \mathrm{lb}=453.6 \mathrm{~g})\) ?

The density of platinum (Pt) is \(21.5 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). What is the volume of \(87.6 \mathrm{~g}\) of \(\mathrm{Pt}\) at this temperature?
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