Solving Equations with Rational Exponents?

Question

The question is 5a^3/4 + 8 = 48 Subtracting 8 than i did this (($$(4\sqrt{5a})^ = 40 $$ and than you get (5A)^3 = (40)^4 i will than get (5a)^3 = 2560000 and when i go and cube root it i will get a decimal which is not the answer. Can anyone help me?

Answer 1

Notice, given that $$5a^{3/4}+8=48$$ $$5a^{3/4}=48-8=40$$

$$5a^{3/4}=40$$ Now, making power $4$ on both the sides, we get $$(5a^{3/4})^4=(40)^4$$ $$(5)^4a^{3}=(40)^4$$ $$\implies a^{3}=\frac{(40)^4}{5^4}$$ $$a^3=\left(\frac{40}{5}\right)^4$$ $$a=\left(\frac{40}{5}\right)^{4/3}=8^{4/3}=16$$

Answer 2

$\begin{array}{rll} 5\cdot a^{\frac{3}{4}} + 8 &= 48&\text{subtract 8 from both sides}\\ 5\cdot a^{\frac{3}{4}}&=40&\text{divide both sides by 5}\\ a^{\frac{3}{4}}&=8&\text{continue simplifying}\end{array}$

Your mistake was $(5)\cdot(a^\frac{3}{4})= 5a^\frac{3}{4} \neq (5a)^\frac{3}{4}$