Arrange the following mathematical expressions in order from least to greatest value:

The correct order from least to greatest value is

(2.5 \times 10^1)^2

, followed by

\sqrt[3]{3.43 \times 10^8}

, then

\sqrt{6.4 \times 10^5}

, and finally

\frac{5.4 \times 10^6}{6 \times 10^3}

Each expression is simplified to a standard integer representation:

(2.5 \times 10^1)^2 = 625

\sqrt[3]{3.43 \times 10^8} = 700

\sqrt{6.4 \times 10^5} = 800

, and

\frac{5.4 \times 10^6}{6 \times 10^3} = 900

. Comparing these integers from smallest to largest yields the correct sequence.

Evaluate the expression

\sqrt{6.4 \times 10^5}

by converting the radicand to a standard square value.

\sqrt{64 \times 10^4} = 8 \times 10^2 = 800

Converting the base to 64 and reducing the exponent of 10 to an even number allows easy extraction of the square root.

Evaluate the expression

(2.5 \times 10^1)^2

by simplifying the terms inside the parentheses first.

(25)^2 = 625

Multiplying 2.5 by 10 simplifies the base to 25 before applying the exponent.

Evaluate the expression

\frac{5.4 \times 10^6}{6 \times 10^3}

using rules of scientific division.

0.9 \times 10^3 = 900

Dividing the coefficients (

5.4 \div 6 = 0.9

) and subtracting the exponent in the denominator from the numerator (

10^{6-3} = 10^3

) yields the result.

Evaluate the expression

\sqrt[3]{3.43 \times 10^8}

by converting the radicand to a standard cube value.

\sqrt[3]{343 \times 10^6} = 7 \times 10^2 = 700

Converting the base to 343 and adjusting the exponent to 6 (a multiple of 3) allows easy extraction of the cube root.

Compare the evaluated integers to determine the correct order from least to greatest.

625 < 700 < 800 < 900

Comparing the simplified values reveals that the correct sequence is

(2.5 \times 10^1)^2

\sqrt[3]{3.43 \times 10^8}

\sqrt{6.4 \times 10^5}

, and

\frac{5.4 \times 10^6}{6 \times 10^3}

Simplification of exponential, radical, and scientific notation expressions to comparable decimal values.

Instead of converting all expressions to standard numbers, convert them all to scientific notation with a base power of

10^2

(i.e.,

6.25 \times 10^2

7.0 \times 10^2

8.0 \times 10^2

, and

9.0 \times 10^2

) to compare their coefficients directly.

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