Questions: 2(a^-1+b^-1)^-1=2ab/(a+b)

Solution Steps

To determine if the given equation is true, we need to simplify the left-hand side and see if it matches the right-hand side. The left-hand side involves the expression \(2\left(a^{-1}+b^{-1}\right)^{-1}\). We can simplify this by finding a common denominator for \(a^{-1}\) and \(b^{-1}\), and then taking the reciprocal. Finally, we compare the simplified expression to the right-hand side \(\frac{2ab}{a+b}\).

We start with the left-hand side of the equation:

\[ 2\left(a^{-1}+b^{-1}\right)^{-1} \]

We can rewrite \(a^{-1}\) and \(b^{-1}\) as \(\frac{1}{a}\) and \(\frac{1}{b}\), respectively. Thus, we have:

\[ 2\left(\frac{1}{a} + \frac{1}{b}\right)^{-1} = 2\left(\frac{b + a}{ab}\right)^{-1} \]

Taking the reciprocal gives us:

\[ 2 \cdot \frac{ab}{a + b} = \frac{2ab}{a + b} \]

The right-hand side of the equation is:

\[ \frac{2ab}{a + b} \]

Now we compare the simplified left-hand side with the right-hand side:

\[ \frac{2ab}{a + b} = \frac{2ab}{a + b} \]

Since both sides are equal, we conclude that the original equation holds true.

Final Answer