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Chapter 5: Problem 40

Write each number without exponents. $$ -9.6 \times 10^{6} $$

Short Answer

Expert verified
The number without exponents is ewlineewline $$-9600000.$$

Step by step solution

01

Understand Scientific Notation

Scientific notation expresses numbers as a product of a number (between 1 and 10) and a power of 10. The given number is ewlineewline $$-9.6 \times 10^6.$$
02

Identify the Base and Exponent

In ewlineewline $$-9.6 \times 10^6,$$ewlineewline the base is ewlineewline $$-9.6$$ ewlineewline and the exponent is ewlineewline $$6.$$
03

Shift the Decimal Point

To rewrite the number without exponents, shift the decimal point to the right by the number of places equal to the exponent (6 in this case).ewlineewline $$-9.6 \times 10^6 \rightarrow -9600000$$.

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Key Concepts

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

Base and Exponent
In scientific notation, numbers are expressed in the form of a base multiplied by a power of 10. For example, in \(-9.6 \times 10^6\), the base is \(-9.6\) and the exponent is \(6\). The base is the number that is being multiplied, and the exponent signifies how many times the base of 10 should be used in the multiplication. Essentially, the notation tells us to multiply \(-9.6\) by \(10^6\), which results in moving the decimal point 6 places to the right. This method helps in representing very large or very small numbers compactly.
Decimal Point Shift
To convert scientific notation back into a standard number, we need to shift the decimal point. The direction and number of shifts depend on the exponent of 10. If the exponent is positive, we shift the decimal point to the right; if it is negative, we shift it to the left. For instance, in the number \(-9.6 \times 10^6\), the exponent is \(6\). Thus, we move the decimal point 6 places to the right. Starting from \(-9.6\), moving the decimal point six places yields \-9600000\. Leading zeros are added for each place the decimal moves over number of digits. This is a systematic way to handle large or tiny amounts efficiently.
Powers of 10
Powers of 10 are fundamental in scientific notation. They determine how many times 10 is multiplied by itself. For example, \(10^6\) means that 10 is used as a factor six times, which equals to 1,000,000. Using powers of 10 simplifies the expression of large and small numbers, making them easier to read and write. This is especially useful in scientific calculations where extremely large quantities or very small measurements, like the mass of an electron, are frequent. Understanding powers of 10 is crucial to mastering scientific notation and correctly shifting the decimal point when converting back into standard numerical format.

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Most popular questions from this chapter

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ \left(2 x+\frac{2}{3} y\right)\left(3 x-\frac{3}{4} y\right) $$

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) See Example 1. $$ 9^{0} $$

Evaluate. $$ 10,000(36.94) $$

Evaluate each expression for \(x=3\) $$ -4 x^{3}+2 x^{2}-9 x-2 $$

In Objective \(I,\) we showed how \(6^{\circ}\) acts as 1 when it is applied to the product rule, thus motivating the definition of 0 as an exponent. We can also use the quotient rule to motivate this definition. Because \(25=5^{2},\) the expression \(\frac{25}{25}\) can be written as the quotient of powers of \(5 .\) Write the expression in this way.
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