Question
How can 13×19 be rewritten using the difference of two squares identity?(1 point) Responses (10+3)(22−3) left parenthesis 10 plus 3 right parenthesis left parenthesis 22 minus 3 right parenthesis (13−3)(19+3) left parenthesis 13 minus 3 right parenthesis left parenthesis 19 plus 3 right parenthesis (16−3)(16+3) left parenthesis 16 minus 3 right parenthesis left parenthesis 16 plus 3 right parenthesis (11−3)(11+3) left parenthesis 11 minus 3 right parenthesis left parenthesis 11 plus 3 right parenthesis
Answers
GPT-4o mini
To rewrite \( 13 \times 19 \) using the difference of two squares identity, we can use the fact that \( a \times b \) can be expressed as \( \left(\frac{a+b}{2}\right)^2 - \left(\frac{a-b}{2}\right)^2 \).
For \( 13 \) and \( 19 \):
- The sum \( a + b = 13 + 19 = 32 \)
- The difference \( a - b = 19 - 13 = 6 \)
Next, we calculate:
- The average: \( \frac{13 + 19}{2} = \frac{32}{2} = 16 \)
- Half the difference: \( \frac{19 - 13}{2} = \frac{6}{2} = 3 \)
Using the difference of squares identity:
\[
13 \times 19 = \left(16 - 3\right)\left(16 + 3\right)
\]
This gives us:
\[
(16 - 3)(16 + 3)
\]
Thus, this expression is valid and can be selected from your options.
The correct response is:
\((16 - 3)(16 + 3)\)
For \( 13 \) and \( 19 \):
- The sum \( a + b = 13 + 19 = 32 \)
- The difference \( a - b = 19 - 13 = 6 \)
Next, we calculate:
- The average: \( \frac{13 + 19}{2} = \frac{32}{2} = 16 \)
- Half the difference: \( \frac{19 - 13}{2} = \frac{6}{2} = 3 \)
Using the difference of squares identity:
\[
13 \times 19 = \left(16 - 3\right)\left(16 + 3\right)
\]
This gives us:
\[
(16 - 3)(16 + 3)
\]
Thus, this expression is valid and can be selected from your options.
The correct response is:
\((16 - 3)(16 + 3)\)