Question

bethany wants to estimate the value of \\((4.296 \times 10^{11})(1.8614 \times 10^{-14})\\). which statement about the estimate is true?

• the value will be greater than 1 because \\(4.296 \times 10^{11}\\) is a very large number and multiplication always increases the size of a number.
• the value will be greater than 1 because \\(4 \times 2 = 8\\) and 8 is larger than 1.
• the value will be less than 1 because \\((4.296 \times 10^{11})(1.8614 \times 10^{-14})\\) will be a negative number. all negative numbers are less than one.
• the number will be less than 1 because when adding the exponents, \\(11 + (-14) = -3\\), and a number in scientific notation with a negative exponent is less than 1.

Explanation:

Estimate the product of the coefficients

$$ 4.296 \times 1.8614 \approx 4 \times 2 = 8 $$

Add the exponents of the powers of ten

$$ 10^{11} \times 10^{-14} = 10^{11 + (-14)} = 10^{-3} $$

Combine the estimates and evaluate

$$ 8 \times 10^{-3} = 0.008 < 1 $$

Answer:

• (A) The value will be greater than 1 because \(4.296 \times 10^{11}\) is a very large number and multiplication always increases the size of a number.
• (B) The value will be greater than 1 because \(4 \times 2 = 8\) and 8 is larger than 1.

<mcq-option>(C) The value will be less than 1 because \((4.296 \times 10^{11})(1.8614 \times 10^{-14})\) will be a negative number. All negative numbers are less than one.</mcq-correct>

• (D) The number will be less than 1 because when adding the exponents, \(11 + (-14) = -3\), and a number in scientific notation with a negative exponent is less than 1. (Correct answer)