The mass of a sample in a laboratory experiment is given by the expression M = 0.000375 \t… — GRE General Test

The mass of a sample in a laboratory experiment is given by the expression $M = 0.000375 \times 10^n$ grams, where $n$ is an integer. When $M$ is written in standard scientific notation as $a \times 10^k$ , where $1 \le a < 10$ and $k$ is an integer, the exponent $k$ is equal to $-2$ . What is the value of $n$ ?

A
$-6$
B
$-2$
$2$ Answer
D
$4$
E
$6$

Answer

The value of

n

2

Writing

0.000375

3.75 \times 10^{-4}

allows the mass

M

to be rewritten as

3.75 \times 10^{-4 + n}

. Matching this with the standard scientific notation form

a \times 10^k

(where

a = 3.75

and

k = -2

) gives the equation

-4 + n = -2

. Solving for

n

yields

n = 2

Step-by-Step Solution

Convert the decimal decimal coefficient to scientific notation.

0.000375 = 3.75 \times 10^{-4}

Moving the decimal point 4 places to the right puts the leading number in the required range

1 \le a < 10

Substitute this conversion back into the expression for

M

and combine powers of 10.

M = (3.75 \times 10^{-4}) \times 10^n = 3.75 \times 10^{n - 4}

By exponent rules,

10^a \times 10^b = 10^{a+b}

Equate the exponent of 10 in the simplified expression to the given value of

k

n - 4 = -2 \implies n = 2

The scientific notation requires

a \times 10^k

where

k = -2

, so

n - 4

must equal

-2

Key Concept

Converting numbers between standard decimal form and scientific notation using place value and exponent properties

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