Question

6.2 scientific notation quiz

what must be true before you can add or subtract numbers in scientific notation?

both the coefficients and exponents must be 10.
the coefficients must be equal.
the exponents must be equal.
you must multiply them first.

Explanation:

Analyze the requirements for adding or subtracting

To add or subtract numbers written in scientific notation, we must align their place values. This is mathematically equivalent to factoring out a common power of 10. Therefore, the powers of 10 (the exponents) must be identical before we can add or subtract the coefficients.

Evaluate the given options

• Option 1: "Both the coefficients and exponents must be 10." This is incorrect because coefficients can be any real number \(a\) where \(1 \le |a| < 10\).
• Option 2: "The coefficients must be equal." This is incorrect because we can add or subtract different coefficients once their place values are aligned.
• Option 3: "The exponents must be equal." This is correct because equal exponents ensure that the coefficients represent the same place value, allowing direct addition or subtraction.
• Option 4: "You must multiply them first." This is incorrect because multiplication is a completely different operation and is not a prerequisite for addition or subtraction.

Connect to prior knowledge

Using the Scientific Notation Multiplication knowledge point, we recall that when multiplying, we multiply the coefficients and add the exponents:

$$ (a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n} $$

However, for addition and subtraction, we rely on the distributive property:

$$ (a \times 10^n) \pm (b \times 10^n) = (a \pm b) \times 10^n $$

This factorization is only possible when the exponents are equal.

Answer:

• Both the coefficients and exponents must be 10.
• The coefficients must be equal.
• The exponents must be equal. (Correct answer)
• You must multiply them first.