Question

which expression is equivalent to $2q^{\frac{3}{4}}$?\
$\sqrt4{2q^{3}}$ $\sqrt3{2q^{4}}$ $\frac{2\sqrt3{q}}{\sqrt4{q}}$ $2\sqrt4{q^{3}}$

Explanation:

Step1: Recall the rule for rational exponents

The formula to convert a rational exponent \(a^{\frac{m}{n}}\) to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (or equivalently \((\sqrt[n]{a})^{m}\)). For the term \(q^{\frac{3}{4}}\), using this rule, we can rewrite it as \(\sqrt[4]{q^{3}}\) (since \(n = 4\) and \(m=3\)). The coefficient \(2\) remains as it is. So, \(2q^{\frac{3}{4}}=2\times\sqrt[4]{q^{3}} = 2\sqrt[4]{q^{3}}\).

Step2: Analyze each option

• Option 1: \(\sqrt[4]{2q^{3}}\) is not equivalent because the coefficient \(2\) is inside the radical, while in our expression the coefficient \(2\) is outside the radical.
• Option 2: \(\sqrt[3]{2q^{4}}\) has a cube root and the exponent of \(q\) is \(4\), which does not match the rational exponent form we need (we need a fourth root and exponent of \(q\) as \(3\)).
• Option 3: \(\frac{2\sqrt[3]{q}}{\sqrt[4]{q}}\) involves different roots and a fraction, which is not equivalent to \(2q^{\frac{3}{4}}\).
• Option 4: \(2\sqrt[4]{q^{3}}\) matches our derived expression from Step 1.

Answer:

\(2\sqrt[4]{q^{3}}\) (the fourth option: \(2\sqrt[4]{q^{3}}\))