Select the correct answer.
What is the simplified form of [tex]$3 \sqrt{7}-5 \sqrt{7}$[/tex]?
A. [tex]$-2 \sqrt{7}$[/tex]
B. [tex]$2 \sqrt{7}$[/tex]
C. [tex]$-\sqrt{14}$[/tex]
D. [tex]$15 \sqrt{7}$[/tex]
Answer (2)
Factor out the common radical: 3 7 − 5 7 = ( 3 − 5 ) 7 .
Simplify the expression inside the parentheses: 3 − 5 = − 2 .
Substitute the simplified value back into the expression: ( 3 − 5 ) 7 = − 2 7 .
The simplified form is − 2 7 .
Explanation
Understanding the Problem We are asked to simplify the expression 3 7 − 5 7 . Both terms contain the same radical, 7 , so we can combine them.
Factoring out the common factor To simplify the expression, we factor out the common factor 7 from both terms: 3 7 − 5 7 = ( 3 − 5 ) 7
Simplifying the expression inside the parentheses Now, we simplify the expression inside the parentheses: 3 − 5 = − 2
Final Answer Substitute the simplified value back into the expression: ( 3 − 5 ) 7 = − 2 7 Therefore, the simplified form of 3 7 − 5 7 is − 2 7 .
Examples
Radicals are useful in many fields, including engineering and physics. For example, when calculating the impedance of an AC circuit, you often encounter expressions involving square roots. Simplifying these expressions helps in determining the overall behavior of the circuit. Also, radicals appear when calculating distances using the Pythagorean theorem, such as finding the length of a diagonal brace in construction or determining the shortest path in navigation.
The simplified form of 3 7 − 5 7 is − 2 7 . Therefore, the correct answer is option A: − 2 7 .
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