Review question

# Given these power facts, can we show $z^6=cx$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9160

## Solution

If $x = ay^{\frac{3}{4}}$ and $z = b \sqrt{(y/x)}$, prove that $z^6 = cx$, where $c$ is independent of $x$, $y$ and $z$, stating the value of $c$ in terms of $a$ and $b$.

The rules of indices include

$(pq)^r = p^rq^r$ $(p^q)^r = p^{qr}$ $p^qp^r = p^{q+r}$

We will apply these rules repeatedly here. We have that

$z = b \sqrt{y/x} \implies z = b x^{-1/2}y^{1/2} \implies z^6 = b^6y^3x^{-3}.$

We also have that

$x = ay^{3/4} \implies x^4=a^4y^3 \implies y^3 = x^4a^{-4}.$

So $z^6 = b^6a^{-4}x^4x^{-3} = b^6a^{-4}x$, and so $c = \dfrac{b^6}{a^4}.$