Question

aliyah wants to estimate the value of the quotient of (5.68 \times 10^{-3}) and (8.54 \times 10^{-9}). which statement about the value is true?

• the value will be greater than 1 because ((-3) \div (-9)) is a positive number. a number in scientific notation with a positive exponent has a value greater than 1.
• the value will be greater than 1 because ((-3) - (-9) = 6). a number in scientific notation with a positive exponent is greater than 1.
• the value will be less than 1 because (5.68 \div 8.54) is less than 1. the negative exponents will make it even smaller.
• the number will be less than 1 because (5.68 \times 10^{-3}) is a very small number and division always makes the size of a number smaller.

Explanation:

Set up the quotient expression

$$ Q = \frac{5.68 \times 10^{-3}}{8.54 \times 10^{-9}} $$

Apply exponent rules to the powers of ten

$$ \frac{10^{-3}}{10^{-9}} = 10^{-3 - (-9)} = 10^{-3 + 9} = 10^6 $$

Estimate the final value

$$ Q = \frac{5.68}{8.54} \times 10^6 \approx 0.665 \times 10^6 = 6.65 \times 10^5 $$

$$ 6.65 \times 10^5 > 1 $$

Answer:

• The value will be greater than 1 because \((-3) \div (-9)\) is a positive number. A number in scientific notation with a positive exponent has a value greater than 1.
• The value will be greater than 1 because \((-3) - (-9) = 6\). A number in scientific notation with a positive exponent is greater than 1. (Correct answer)
• The value will be less than 1 because \(5.68 \div 8.54\) is less than 1. The negative exponents will make it even smaller.
• The number will be less than 1 because \(5.68 \times 10^{-3}\) is a very small number and division always makes the size of a number smaller.