Simplify the Expression: m²/9 - 4mn/3 + 4n² Step-by-Step

Question

m2943mn+4n2=? \frac{m^2}{9}-\frac{4}{3}mn+4n^2=\text{?}

Video Solution

Solution Steps

00:00 Express as brackets
00:03 Break down 9 into 3 squared
00:11 Break down 4 into 2 squared
00:17 Extract root and square
00:29 Factor with 2 and 2
00:32 Extract root and square
00:49 Use shortened multiplication formulas to find the brackets
01:00 These are A and B
01:06 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we'll simplify the expression m2943mn+4n2\frac{m^2}{9} - \frac{4}{3}mn + 4n^2 to reveal a possible square of a binomial:

  • Step 1: Recognize the expression m2943mn+4n2\frac{m^2}{9} - \frac{4}{3}mn + 4n^2 may have terms fitting the identity (ab)2=a22ab+b2(a - b)^2 = a^2 - 2ab + b^2.
  • Step 2: Identify parts of the expression:
    - a2=m29a^2 = \frac{m^2}{9}, so a=m3a = \frac{m}{3}.
    - 2ab=43mn-2ab = -\frac{4}{3}mn. Since a=m3a = \frac{m}{3}, solve for bb:
    2×m3×b=43mn2b=4nb=2n2 \times \frac{m}{3} \times b = \frac{4}{3}mn \Rightarrow 2b = 4n \Rightarrow b = 2n.
  • Step 3: Check b2b^2: b2=(2n)2=4n2b^2 = (2n)^2 = 4n^2, which matches the 4n24n^2 term in the expression.

Therefore, the correct form of the expression is:
(m32n)2(\frac{m}{3} - 2n)^2.

Thus, the expression simplifies to:

(m32n)2 (\frac{m}{3} - 2n)^2

Answer

(m32n)2 (\frac{m}{3}-2n)^2