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Difficulty: MediumExponents, Roots, and Scientific Notation

The variable pp represents the value 8×1058 \times 10^5, and the variable qq represents the value 2×10−32 \times 10^{-3}. When the expression pq\sqrt{\frac{p}{q}} is simplified and written in scientific notation, which of the following is the resulting value?

  1. A
    2×1012 \times 10^1
  2. 2×1042 \times 10^4Answer
  3. C
    2×1082 \times 10^8
  4. D
    4×1044 \times 10^4
  5. E
    4×1084 \times 10^8

Answer

2×1042 \times 10^4
The correct answer is obtained by first simplifying the quotient inside the radical: 8×1052×10−3=4×105−(−3)=4×108\frac{8 \times 10^5}{2 \times 10^{-3}} = 4 \times 10^{5 - (-3)} = 4 \times 10^8. Taking the square root of this value yields 4×108=2×104\sqrt{4} \times \sqrt{10^8} = 2 \times 10^4, which is written in standard scientific notation.

Step-by-Step Solution

1
Substitute the values of pp and qq into the fraction inside the radical.
8×1052×10−3\frac{8 \times 10^5}{2 \times 10^{-3}}
To evaluate the quotient of the two numbers before applying the root.
2
Divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator.
82×105−(−3)=4×108\frac{8}{2} \times 10^{5 - (-3)} = 4 \times 10^8
Using the quotient rule for exponents: 10a10b=10a−b\frac{10^a}{10^b} = 10^{a-b}.
3
Apply the square root to both the coefficient and the power of 10.
4×108=2×108×12=2×104\sqrt{4} \times \sqrt{10^8} = 2 \times 10^{8 \times \frac{1}{2}} = 2 \times 10^4
Using the properties of radicals: ab=ab\sqrt{ab} = \sqrt{a}\sqrt{b} and 102k=10k\sqrt{10^{2k}} = 10^k.

Key Concept

Exponents, Roots, and Scientific Notation
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Topics

Exponents, Roots, and Scientific Notation

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