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Chapter 1: Problem 89

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\left(7.2 \times 10^{-9}\right)\left(1.806 \times 10^{-12}\right)$$

Short Answer

Expert verified
\(1.3 \times 10^{-20}\)

Step by step solution

01

Rewrite Numbers in Standard Form

First, express each number in regular decimal notation. We have:\(7.2 \times 10^{-9}\) which is \(0.0000000072\), and \(1.806 \times 10^{-12}\) which is \(0.000000000001806\).
02

Multiply Coefficients

Next, we'll multiply the coefficients of the two numbers: \(7.2 \times 1.806 = 12.9912\).
03

Use Laws of Exponents for Powers of Ten

We multiply the powers of ten separately by adding their exponents: \(10^{-9} \times 10^{-12} = 10^{-21}\).
04

Combine Results

Now, we combine our results from Step 2 and Step 3 into one expression: \(12.9912 \times 10^{-21}\).
05

Adjust to Scientific Notation

Adjust the number to proper scientific notation. This requires converting \(12.9912\) to \(1.29912 \times 10^1\). Thus, our expression becomes:\(1.29912 \times 10^1 \times 10^{-21} = 1.29912 \times 10^{-20}\).
06

Correct for Significant Digits

The initial terms had 2 and 4 significant digits, respectively. The result should have the same as the smaller number, which is 2. So the final answer is rounded to \(1.3 \times 10^{-20}\).

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Key Concepts

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

Laws of Exponents
The laws of exponents are fundamental rules that simplify calculations involving powers of numbers. These laws make it easier to handle complex mathematical operations especially in scientific notation, where very large or very small numbers are expressed as powers of ten.

Here are some critical rules to remember:
  • Multiplication of Powers: When multiplying numbers with the same base, you add their exponents. For example, if you have \(a^m \times a^n\), the result is \(a^{m+n}\).
  • Division of Powers: Division means subtracting the exponents: \(a^m / a^n = a^{m-n}\).
  • Powers of a Power: If you raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
In the provided solution, the multiplication of powers of ten is tackled using "Multiplication of Powers". The exponents \(-9\) and \(-12\) were added to give \(-21\), reflecting \(10^{-9} \times 10^{-12} = 10^{-21}\). Understanding these rules helps to simplify and solve such expressions with ease.
Significant Digits
Significant digits, or significant figures, are the numbers that carry meaning contributing to a measurement's precision. They help communicate how precise a number is, especially in scientific experiments and engineering where precision is critical.

The rules to determine significant digits are:
  • Non-zero digits are always significant.
  • Any zeros between significant digits are also significant.
  • Leading zeros are not significant since they are only placeholders.
  • Trailing zeros in a decimal number are significant because they indicate precision.
In our solution, notice how significant digits guide the rounding process. The expression \(7.2 \times 10^{-9}\) has 2 significant digits, while \(1.806 \times 10^{-12}\) has 4. Thus, when expressing the final result in scientific notation, it is crucial to round it to 2 significant digits to maintain consistency with the initial number having the least precision.
Decimal Notation
Decimal notation is the standard form of writing numbers using the base-ten numeral system. This system is familiar as it reflects how we usually write numbers in everyday life.

Understanding decimal notation is crucial because it simplifies converting between scientific notation and a format that's easy to comprehend.

Let's break it down:
  • In scientific notation, a number is typically of the form \(a \times 10^n\), where \(1 \le a < 10\) and \(n\) is an integer.
  • Converting back to decimal form involves moving the decimal point based on the exponent \(n\). A positive \(n\) moves the decimal point to the right, indicating larger numbers. A negative \(n\) shifts it to the left, representing smaller numbers.
In the exercise, converting \(7.2 \times 10^{-9}\) to \(0.0000000072\) and \(1.806 \times 10^{-12}\) to \(0.000000000001806\) demonstrates moving the decimal point to properly describe the quantities in an easily readable form. Grasping this concept helps in applying scientific notation effectively.

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Most popular questions from this chapter

Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius.

Using Distances to Solve Absolute Value Inequalities \(\quad\) Recall that \(|a-b|\) is the distance between \(a\) and \(b\) on the number line. For any number \(x\) what do \(|x-1|\) and \(|x-3|\) represent? Use this interpretation to solve the inequality \(|x-1|<|x-3|\) geometrically. In general, if \(a \leq b,\) what is the solution of the inequality \(|x-a|<|x-b| ?\)

As of July 2013 , the population of the United States was \(3.164 \times 10^{8},\) and the national debt was \(1.674 \times 10^{13}\) dollars. How much was each person's share of the debt?

DISCUSS: Proof That \(0=1 ?\) The following steps appear to give equivalent equations, which seem to prove that \(1=0 .\) Find the error. \(\begin{aligned} x &=1 & & \text { Given } \\ x^{2} &=x & & \text { Multiply by } x \\ x^{2}-x &=0 & & \text { Subtract } x \end{aligned}$$(x-1)=0\)Factor \(\frac{(x-1)}{x-1}=\frac{0}{x-1} \quad\) Divide by \(x-1$$x=0 \quad\) Simplify\(1=0 \quad\) Given \(x=1\)

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt{4 s t^{3}} \sqrt[6]{s^{3} t^{2}}\) (b) \(\frac{\sqrt[4]{x^{7}}}{\sqrt[4]{x^{3}}}\)
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