Answer :
We can express each number as a multiple of a power of 10 by writing it in scientific notation. This means rewriting the number in the form
[tex]$$
a \times 10^n,
$$[/tex]
where [tex]$a$[/tex] is the multiple (with [tex]$1 \le a < 10$[/tex]) and [tex]$n$[/tex] is an integer. Here is a step-by-step process for each part:
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a. 42,300
1. Write [tex]$42,\!300$[/tex] in standard form by locating the decimal point. The number can be thought of as [tex]$42300.0$[/tex].
2. To have one digit before the decimal point, we shift it 4 places to the left, obtaining [tex]$4.23$[/tex].
3. Since we shifted the decimal point 4 places, we multiply by [tex]$10^4$[/tex].
Thus,
[tex]$$
42,\!300 = 4.23 \times 10^4.
$$[/tex]
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b. 2,000
1. Write [tex]$2,\!000$[/tex] as [tex]$2000.0$[/tex].
2. Shifting the decimal point 3 places to the left gives [tex]$2.0$[/tex], or simply [tex]$2$[/tex].
3. The shift indicates a factor of [tex]$10^3$[/tex].
So,
[tex]$$
2,\!000 = 2 \times 10^3.
$$[/tex]
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c. 9,200,000
1. Express [tex]$9,\!200,\!000$[/tex] as [tex]$9200000.0$[/tex].
2. Moving the decimal point 6 places to the left results in [tex]$9.2$[/tex].
3. Therefore, the power of 10 required is [tex]$10^6$[/tex].
Thus,
[tex]$$
9,\!200,\!000 = 9.2 \times 10^6.
$$[/tex]
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d. Four thousand
1. The phrase "Four thousand" corresponds to the numeral [tex]$4000$[/tex].
2. Writing [tex]$4000$[/tex] as [tex]$4000.0$[/tex], shifting the decimal point 3 places to the left gives [tex]$4.0$[/tex] (or [tex]$4$[/tex]).
3. This yields a multiplication by [tex]$10^3$[/tex].
Hence,
[tex]$$
4000 = 4 \times 10^3.
$$[/tex]
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e. 80 million
1. "80 million" means [tex]$80,\!000,\!000$[/tex].
2. When written as [tex]$80000000.0$[/tex], shifting the decimal 7 places to the left gives [tex]$8.0$[/tex] (or [tex]$8$[/tex]).
3. Therefore, the corresponding power of 10 is [tex]$10^7$[/tex].
Thus,
[tex]$$
80,\!000,\!000 = 8 \times 10^7.
$$[/tex]
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f. 32 billion
1. "32 billion" stands for [tex]$32,\!000,\!000,\!000$[/tex].
2. Written as [tex]$32000000000.0$[/tex], moving the decimal 10 places to the left produces [tex]$3.2$[/tex].
3. This corresponds to multiplying by [tex]$10^{10}$[/tex].
So,
[tex]$$
32,\!000,\!000,\!000 = 3.2 \times 10^{10}.
$$[/tex]
----------------------------------------------------------------
In summary, the expressions as multiples of powers of 10 are:
[tex]\[
\begin{aligned}
a. \quad 42,\!300 &= 4.23 \times 10^4, \\
b. \quad 2,\!000 &= 2 \times 10^3, \\
c. \quad 9,\!200,\!000 &= 9.2 \times 10^6, \\
d. \quad \text{Four thousand} &= 4 \times 10^3, \\
e. \quad 80\text{ million} &= 8 \times 10^7, \\
f. \quad 32\text{ billion} &= 3.2 \times 10^{10}.
\end{aligned}
\][/tex]
Each expression has been rewritten in the required form by determining the appropriate multiple [tex]$a$[/tex] and the exponent [tex]$n$[/tex] such that the original number equals [tex]$a \times 10^n$[/tex].
[tex]$$
a \times 10^n,
$$[/tex]
where [tex]$a$[/tex] is the multiple (with [tex]$1 \le a < 10$[/tex]) and [tex]$n$[/tex] is an integer. Here is a step-by-step process for each part:
----------------------------------------------------------------
a. 42,300
1. Write [tex]$42,\!300$[/tex] in standard form by locating the decimal point. The number can be thought of as [tex]$42300.0$[/tex].
2. To have one digit before the decimal point, we shift it 4 places to the left, obtaining [tex]$4.23$[/tex].
3. Since we shifted the decimal point 4 places, we multiply by [tex]$10^4$[/tex].
Thus,
[tex]$$
42,\!300 = 4.23 \times 10^4.
$$[/tex]
----------------------------------------------------------------
b. 2,000
1. Write [tex]$2,\!000$[/tex] as [tex]$2000.0$[/tex].
2. Shifting the decimal point 3 places to the left gives [tex]$2.0$[/tex], or simply [tex]$2$[/tex].
3. The shift indicates a factor of [tex]$10^3$[/tex].
So,
[tex]$$
2,\!000 = 2 \times 10^3.
$$[/tex]
----------------------------------------------------------------
c. 9,200,000
1. Express [tex]$9,\!200,\!000$[/tex] as [tex]$9200000.0$[/tex].
2. Moving the decimal point 6 places to the left results in [tex]$9.2$[/tex].
3. Therefore, the power of 10 required is [tex]$10^6$[/tex].
Thus,
[tex]$$
9,\!200,\!000 = 9.2 \times 10^6.
$$[/tex]
----------------------------------------------------------------
d. Four thousand
1. The phrase "Four thousand" corresponds to the numeral [tex]$4000$[/tex].
2. Writing [tex]$4000$[/tex] as [tex]$4000.0$[/tex], shifting the decimal point 3 places to the left gives [tex]$4.0$[/tex] (or [tex]$4$[/tex]).
3. This yields a multiplication by [tex]$10^3$[/tex].
Hence,
[tex]$$
4000 = 4 \times 10^3.
$$[/tex]
----------------------------------------------------------------
e. 80 million
1. "80 million" means [tex]$80,\!000,\!000$[/tex].
2. When written as [tex]$80000000.0$[/tex], shifting the decimal 7 places to the left gives [tex]$8.0$[/tex] (or [tex]$8$[/tex]).
3. Therefore, the corresponding power of 10 is [tex]$10^7$[/tex].
Thus,
[tex]$$
80,\!000,\!000 = 8 \times 10^7.
$$[/tex]
----------------------------------------------------------------
f. 32 billion
1. "32 billion" stands for [tex]$32,\!000,\!000,\!000$[/tex].
2. Written as [tex]$32000000000.0$[/tex], moving the decimal 10 places to the left produces [tex]$3.2$[/tex].
3. This corresponds to multiplying by [tex]$10^{10}$[/tex].
So,
[tex]$$
32,\!000,\!000,\!000 = 3.2 \times 10^{10}.
$$[/tex]
----------------------------------------------------------------
In summary, the expressions as multiples of powers of 10 are:
[tex]\[
\begin{aligned}
a. \quad 42,\!300 &= 4.23 \times 10^4, \\
b. \quad 2,\!000 &= 2 \times 10^3, \\
c. \quad 9,\!200,\!000 &= 9.2 \times 10^6, \\
d. \quad \text{Four thousand} &= 4 \times 10^3, \\
e. \quad 80\text{ million} &= 8 \times 10^7, \\
f. \quad 32\text{ billion} &= 3.2 \times 10^{10}.
\end{aligned}
\][/tex]
Each expression has been rewritten in the required form by determining the appropriate multiple [tex]$a$[/tex] and the exponent [tex]$n$[/tex] such that the original number equals [tex]$a \times 10^n$[/tex].
