Question

match the expression ( x^{\frac{5}{9}} ) with the equivalent expression from the choices given below. (sqrt5{x^{9}}) (sqrt9{x^{5}}) (left(sqrt{x^{5}}
ight)^{9}) (left(sqrt{x^{9}}
ight)^{5}) choose the correct answer below. (\bigcirc) (left(sqrt{x^{9}}
ight)^{5}) (\bigcirc) (sqrt5{x^{9}}) (\bigcirc) (sqrt9{x^{5}}) (\bigcirc) (left(sqrt{x^{5}}
ight)^{9})

Explanation:

Step1: Recall the exponent - root relationship

The formula for converting a rational exponent to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (where \(n\) is the index of the radical and \(m\) is the power of the base inside the radical), and also \(a^{\frac{m}{n}} = (\sqrt[n]{a})^{m}\). For the expression \(x^{\frac{5}{9}}\), comparing with \(a^{\frac{m}{n}}\), we have \(a = x\), \(m = 5\) and \(n=9\).

Step2: Apply the formula

Using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), when \(a = x\), \(m = 5\) and \(n = 9\), we get \(x^{\frac{5}{9}}=\sqrt[9]{x^{5}}\). We can also check using the other form \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}\), here \((\sqrt[9]{x})^{5}=\sqrt[9]{x^{5}}\) (since \((\sqrt[n]{a})^{m}=\sqrt[n]{a^{m}}\) by the property of exponents \((a^{b})^{c}=a^{bc}\), so \((\sqrt[9]{x})^{5}=(x^{\frac{1}{9}})^{5}=x^{\frac{5}{9}}\) and \(\sqrt[9]{x^{5}}=(x^{5})^{\frac{1}{9}}=x^{\frac{5}{9}}\)).

Let's check the other options:

• For \(\sqrt[5]{x^{9}}\), using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), this is \(x^{\frac{9}{5}}

eq x^{\frac{5}{9}}\).

• For \((\sqrt{x^{5}})^{9}\), \(\sqrt{x^{5}}=x^{\frac{5}{2}}\), then \((x^{\frac{5}{2}})^{9}=x^{\frac{45}{2}}

eq x^{\frac{5}{9}}\).

• For \((\sqrt{x^{9}})^{5}\), \(\sqrt{x^{9}}=x^{\frac{9}{2}}\), then \((x^{\frac{9}{2}})^{5}=x^{\frac{45}{2}}

eq x^{\frac{5}{9}}\).

Answer:

\(\boldsymbol{\sqrt[9]{x^{5}}}\) (or in the option format: the option with \(\sqrt[9]{x^{5}}\))