Question

Find the constant of variation and the degree. Is it a power function? (See pictures)

Answer 1

The given function is a power function.

`\textsf{The constant of variation is $\sqrt[3]{(7)/(5)}$}.`

`\textsf{The power is $(1)/(3)$}.`

Explanation:

Power function

Any function that can be written in the form:

`\boxed{f(x)=k \cdot x^a}`

where:

• a is the power.
• k is the constant of variation.

Given function:

`f(x)=\sqrt[3]{(7x)/(5)}`

`\textsf{Apply radical rule} \quad \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}:`

`f(x)=\sqrt[3]{(7)/(5)\cdot x}`

`\implies f(x)=\sqrt[3]{(7)/(5)} \cdot \sqrt[3]{x}`

`\textsf{Apply exponent rule} \quad \sqrt[n]{a}=a^{(1)/(n)}:`

`\implies f(x)=\sqrt[3]{(7)/(5)} \cdot x^{(1)/(3)}`

Therefore, the given function is a power function.

`\textsf{The constant of variation is $\sqrt[3]{(7)/(5)}$}.`

`\textsf{The power is $(1)/(3)$}.`