Question

which expression does not simplify to 1?
\\(\frac{g^{3}}{g cdot g cdot g}\\)
\\(\frac{g h^{2}}{h^{2} g}\\)
\\(\frac{g^{4} h^{2}}{g h^{6}}\\)
\\(\frac{g cdot g}{g^{2}}\\)

Explanation:

Step1: Simplify the first expression

The first expression is \(\frac{g^{3}}{g\cdot g\cdot g}\). Since \(g\cdot g\cdot g = g^{3}\), we have \(\frac{g^{3}}{g^{3}} = 1\).

Step2: Simplify the second expression

The second expression is \(\frac{gh^{2}}{h^{2}g}\). By the commutative property of multiplication, the numerator and denominator are the same, so \(\frac{gh^{2}}{h^{2}g}=1\).

Step3: Simplify the third expression

The third expression is \(\frac{g^{4}h^{2}}{gh^{6}}\). Using the quotient rule for exponents (\(\frac{a^{m}}{a^{n}}=a^{m - n}\)), for \(g\) we have \(g^{4-1}=g^{3}\), and for \(h\) we have \(h^{2 - 6}=h^{- 4}=\frac{1}{h^{4}}\). So the expression simplifies to \(g^{3}\cdot\frac{1}{h^{4}}=\frac{g^{3}}{h^{4}}
eq1\).

Step4: Simplify the fourth expression

The fourth expression is \(\frac{g\cdot g}{g^{2}}\). Since \(g\cdot g = g^{2}\), we have \(\frac{g^{2}}{g^{2}} = 1\).

Answer:

The expression \(\boldsymbol{\frac{g^{4}h^{2}}{gh^{6}}}\) does not simplify to 1.