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Difficulty: HardExponents, Roots, and Scientific Notation

Simplify each of the following expressions and arrange them in order of their value from least to greatest.

  1. 14.0×104\sqrt{4.0 \times 10^4}
  2. 2(5.0×103)×(5.0×10−2)(5.0 \times 10^3) \times (5.0 \times 10^{-2})
  3. 34.8×1051.6×103\frac{4.8 \times 10^5}{1.6 \times 10^3}
  4. 46.4×1073\sqrt[3]{6.4 \times 10^7}

Answer

The correct order of the expressions from least to greatest value is: the expression with square root of four times ten to the fourth power (200), the product expression (250), the division expression (300), and finally the cube root expression (400).
Evaluating all four expressions yields the values 200, 400, 300, and 250 respectively. Arranging these values from least to greatest yields 200, 250, 300, and 400.

Step-by-Step Solution

1
Evaluate the first expression: 4.0×104\sqrt{4.0 \times 10^4}.
200200
Using the properties of square roots and exponents, rewrite as 4.0×104=2×102=200\sqrt{4.0} \times \sqrt{10^4} = 2 \times 10^2 = 200.
2
Evaluate the second expression: 6.4×1073\sqrt[3]{6.4 \times 10^7}.
400400
Adjust the scientific notation to make the exponent divisible by 3: 6.4×107=64×1066.4 \times 10^7 = 64 \times 10^6. Then evaluate the cube root: 643×1063=4×102=400\sqrt[3]{64} \times \sqrt[3]{10^6} = 4 \times 10^2 = 400.
3
Evaluate the third expression: 4.8×1051.6×103\frac{4.8 \times 10^5}{1.6 \times 10^3}.
300300
Divide the coefficients 4.81.6=3\frac{4.8}{1.6} = 3 and subtract the exponents in the denominator from the numerator 105−3=10210^{5-3} = 10^2, yielding 3×102=3003 \times 10^2 = 300.
4
Evaluate the fourth expression: (5.0×103)×(5.0×10−2)(5.0 \times 10^3) \times (5.0 \times 10^{-2}).
250250
Multiply the coefficients 5.0×5.0=25.05.0 \times 5.0 = 25.0 and add the exponents 3+(−2)=13 + (-2) = 1. The result is 25.0×101=25025.0 \times 10^1 = 250.
5
Compare the evaluated values to arrange them from least to greatest.
200<250<300<400200 < 250 < 300 < 400
Comparing the values gives the final order: 4.0×104\sqrt{4.0 \times 10^4} (200), followed by (5.0×103)×(5.0×10−2)(5.0 \times 10^3) \times (5.0 \times 10^{-2}) (250), then 4.8×1051.6×103\frac{4.8 \times 10^5}{1.6 \times 10^3} (300), and finally 6.4×1073\sqrt[3]{6.4 \times 10^7} (400).

Key Concept

Applying exponent rules, square and cube root properties, and arithmetic operations on numbers in scientific notation.
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Topics

Exponents, Roots, and Scientific Notation

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