If k = 4.5 \times 10^{-8}, what is the value of \frac{(0.000009)^2 \times 400,000}{k} expr… — GRE General Test

If $k = 4.5 \times 10^{-8}$ , what is the value of $\frac{(0.000009)^2 \times 400,000}{k}$ expressed in scientific notation?

7.2 \times 10^2

Converting

0.000009

9 \times 10^{-6}

and squaring yields

81 \times 10^{-12}

. Multiplying this by

400,000 = 4 \times 10^5

produces

324 \times 10^{-7}

. Dividing by

4.5 \times 10^{-8}

gives

72 \times 10^1

, which in proper scientific notation (

a \times 10^n

with

1 \le a < 10

) is

7.2 \times 10^2

Convert the components of the numerator into scientific notation

0.000009 = 9 \times 10^{-6}

and

400,000 = 4 \times 10^5

Converting all numbers to scientific notation simplifies subsequent exponent operations.

Square the first decimal term

(9 \times 10^{-6})^2 = 9^2 \times (10^{-6})^2 = 81 \times 10^{-12}

Applying the exponent to both the coefficient and the power of 10 gives

81 \times 10^{-12}

Multiply the terms in the numerator

(81 \times 10^{-12}) \times (4 \times 10^5) = (81 \times 4) \times 10^{-12 + 5} = 324 \times 10^{-7}

Multiply coefficients and add powers of 10.

Divide the numerator by the denominator

k = 4.5 \times 10^{-8}

\frac{324 \times 10^{-7}}{4.5 \times 10^{-8}} = \left(\frac{324}{4.5}\right) \times 10^{-7 - (-8)} = 72 \times 10^1

Divide coefficients and subtract the denominator exponent from the numerator exponent.

Convert the final result to standard scientific notation format

a \times 10^n

where

1 \le a < 10

72 \times 10^1 = 7.2 \times 10^2

Shift the decimal point one place to the left and increase the exponent of 10 by 1.

Decimals and Scientific Notation Operations

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