Use the properties of rational exponents to simplify the expression
$(3^\frac{1}3 \cdot 4^\frac{1}{4})^3$
I got $3^1 \cdot 4^\frac{3}{4}$
I just wasn't sure if this was the most simplified expression. It says to do it without a calculator so can someone confirm if I'm right?
On
$$(3^{1/3}\cdot 4^{1/4})^3$$
distribute the exponent using the rule that $(a\cdot b)^c = a^c\cdot b^c$ and the rule that $(a^b)^c = a^{(bc)}$ (which as alluded to are only guaranteed for when dealing with positive numbers). This leads us to:
$$(3^{3/3}\cdot 4^{3/4})$$
Simplify the fraction $3/3$ and recognize that this would leave it with an exponent of $1$ and exponents of $1$ can be omitted for brevity. This leads us to:
$$3\cdot 4^{3/4}$$
This is a perfectly fine place to stop if you like. If you prefer, you can continue on however. Recognizing $4$ as $2^2$, we could have continued as:
$$3\cdot (2^2)^{3/4}$$
$$3\cdot 2^{3/2}$$
and if you prefer writing things with surds
$$3\cdot 2\cdot \sqrt{2}$$
$$6\sqrt{2}$$
$$(3^\frac{1}3 \cdot 4^\frac{1}{4})^3$$ $$=(3^\frac{1}3 \cdot 4^\frac{1}{3}\cdot 4^\frac{-1}{12})^3$$ $$=(12^\frac{1}3\cdot 4^\frac{-1}{12})^3$$ $$=12(4^\frac{-1}{12})^3$$ $$=\frac{12}{(4^\frac{1}{12})^3}$$ $$=\frac{12}{4^\frac{1}{4}}$$ $$=\frac{12}{\sqrt{2}}$$ $$=6\sqrt{2}$$
So after simplifying the expression reduces to $$6\sqrt{2}$$