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Zorluk: Çok zorExponents, Roots, and Scientific Notation

Simplify the four mathematical expressions below. How should these expressions be ordered from least to greatest based on their simplified values?

* Expression A: 2.25×104×(2.0×10−1)−2\sqrt{2.25 \times 10^4} \times \left(2.0 \times 10^{-1}\right)^{-2}
* Expression B: (3.0×102)36.0×103\frac{\left(3.0 \times 10^2\right)^3}{6.0 \times 10^3}
* Expression C: 2.7×1073×(4.0×10−2)−1/2\sqrt[3]{2.7 \times 10^7} \times \left(4.0 \times 10^{-2}\right)^{-1/2}
* Expression D: (6.4×105)1/2×5.0×100\left(6.4 \times 10^5\right)^{1/2} \times 5.0 \times 10^0

  1. 1Expression C
  2. 2Expression A
  3. 3Expression D
  4. 4Expression B

Cevap

Expression C, Expression A, Expression D, Expression B
The correct order is determined by fully simplifying each expression: Expression C simplifies to 1,5001,500, Expression A simplifies to 3,7503,750, Expression D simplifies to 4,0004,000, and Expression B simplifies to 4,5004,500. Arranging these values in ascending order gives the correct sequence starting with Expression C, followed by Expression A, then Expression D, and ending with Expression B.

Adım Adım Çözüm

1
Simplify Expression C: 2.7×1073×(4.0×10−2)−1/2\sqrt[3]{2.7 \times 10^7} \times \left(4.0 \times 10^{-2}\right)^{-1/2}
C=1,500C = 1,500
First, rewrite 2.7×1072.7 \times 10^7 as 27×10627 \times 10^6 to compute the cube root easily: 27×1063=3×102=300\sqrt[3]{27 \times 10^6} = 3 \times 10^2 = 300. Second, evaluate the term with the rational exponent: (4.0×10−2)−1/2=14.0×10−2=12.0×10−1=10.2=5\left(4.0 \times 10^{-2}\right)^{-1/2} = \frac{1}{\sqrt{4.0 \times 10^{-2}}} = \frac{1}{2.0 \times 10^{-1}} = \frac{1}{0.2} = 5. Finally, multiply the two parts: 300×5=1,500300 \times 5 = 1,500.
2
Simplify Expression A: 2.25×104×(2.0×10−1)−2\sqrt{2.25 \times 10^4} \times \left(2.0 \times 10^{-1}\right)^{-2}
A=3,750A = 3,750
First, calculate the square root: 2.25×104=2.25×104=1.5×102=150\sqrt{2.25 \times 10^4} = \sqrt{2.25} \times \sqrt{10^4} = 1.5 \times 10^2 = 150. Second, evaluate the term with the negative exponent: (2.0×10−1)−2=(0.2)−2=10.04=25\left(2.0 \times 10^{-1}\right)^{-2} = (0.2)^{-2} = \frac{1}{0.04} = 25. Finally, multiply the two simplified values: 150×25=3,750150 \times 25 = 3,750.
3
Simplify Expression D: (6.4×105)1/2×5.0×100\left(6.4 \times 10^5\right)^{1/2} \times 5.0 \times 10^0
D=4,000D = 4,000
First, simplify the square root by rewriting the radicand: (6.4×105)1/2=64×104=8×102=800\left(6.4 \times 10^5\right)^{1/2} = \sqrt{64 \times 10^4} = 8 \times 10^2 = 800. Second, evaluate 5.0×100=5.0×1=55.0 \times 10^0 = 5.0 \times 1 = 5. Finally, multiply the results: 800×5=4,000800 \times 5 = 4,000.
4
Simplify Expression B: (3.0×102)36.0×103\frac{\left(3.0 \times 10^2\right)^3}{6.0 \times 10^3}
B=4,500B = 4,500
First, raise the numerator to the power of three: (3.0×102)3=33×(102)3=27×106\left(3.0 \times 10^2\right)^3 = 3^3 \times \left(10^2\right)^3 = 27 \times 10^6. Next, divide by the denominator: 27×1066.0×103=276.0×103=4.5×103=4,500\frac{27 \times 10^6}{6.0 \times 10^3} = \frac{27}{6.0} \times 10^3 = 4.5 \times 10^3 = 4,500.
5
Compare the simplified values of the four expressions.
1,500<3,750<4,000<4,5001,500 < 3,750 < 4,000 < 4,500, which corresponds to Expression C, Expression A, Expression D, then Expression B.
Sorting the values in ascending numerical order reveals the correct order from least to greatest.

Anahtar Kavram

Simplifying complex numeric expressions containing combinations of square roots, cube roots, negative exponents, fractional exponents, and scientific notation.
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Exponents, Roots, and Scientific Notation

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