Arrange the following numbers in order from least to greatest: \left(\frac{1}{4}\right)^3,… — ACT

Arrange the following numbers in order from least to greatest: $\left(\frac{1}{4}\right)^3$ , $\sqrt{0.0004}$ , $2.4 \times 10^{-2}$ , and $5^{-2}$ .

1 $\left(\frac{1}{4}\right)^3$
2 $\sqrt{0.0004}$
3 $2.4 \times 10^{-2}$
4 $5^{-2}$

Answer

The correct order of the values from least to greatest is

\left(\frac{1}{4}\right)^3

, followed by

\sqrt{0.0004}

, then

2.4 \times 10^{-2}

, and finally

5^{-2}

Converting all expressions to decimals allows direct comparison:

\left(\frac{1}{4}\right)^3 = 0.015625

\sqrt{0.0004} = 0.02

2.4 \times 10^{-2} = 0.024

, and

5^{-2} = 0.04

. Arranging these decimals from smallest to largest results in the sequence

0.015625 < 0.02 < 0.024 < 0.04

Step-by-Step Solution

Convert each of the exponential, root, and scientific notation expressions into decimal values to make comparison straightforward.

The expressions convert as follows:
-

\left(\frac{1}{4}\right)^3 = \frac{1}{64} = 0.015625

\sqrt{0.0004} = \sqrt{4 \times 10^{-4}} = 2 \times 10^{-2} = 0.02

2.4 \times 10^{-2} = 0.024

5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04

Expressing all values in standard decimal notation eliminates different bases and representations, allowing direct comparison.

Compare and arrange the resulting decimals from the smallest value to the largest value.

0.015625 < 0.02 < 0.024 < 0.04

By aligning the place values, we can see that

0.015625

(one hundredth plus fractional parts) is less than

0.02

(two hundredths), which is less than

0.024

(twenty-four thousandths), which is less than

0.04

(four hundredths).

Key Concept

Comparing numerical expressions involving negative exponents, fractional powers, roots, and scientific notation by converting them to a common decimal representation.

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