Question

assume g and h are whole numbers, and g < h. which expression has the least value?
\\(\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 \(\frac{gh^{2}}{h^{2}g}\). The numerator and denominator are the same (multiplication is commutative, so \(gh^{2}=h^{2}g\)), 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 \(g\): \(g^{4-1}=g^{3}\), and for \(h\): \(h^{2 - 6}=h^{-4}=\frac{1}{h^{4}}\). So the expression simplifies to \(\frac{g^{3}}{h^{4}}\). Since \(g\) and \(h\) are whole numbers and \(g < h\), \(h\geq g + 1\). Let's take the smallest whole numbers where \(g < h\), say \(g = 1\), \(h = 2\). Then \(\frac{1^{3}}{2^{4}}=\frac{1}{16}\), which is less than 1.

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:

\(\boldsymbol{\frac{g^{4}h^{2}}{gh^{6}}}\) (the third expression)