Simplify $\sqrt{\dfrac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$ into $\dfrac{\sqrt{3}}{3}$

Question

I am on the final question of a textbook chapter on radicals and this question feels more challenging, perhaps that's the idea. If you view my post history I typically make a effort to provide some working to simplify the expression to an extent, but here I am very confused about where to go or my first steps.

I am to simplify:

$$\sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$$

The solution is $\displaystyle \frac{\sqrt{3}}{3}$

The expression makes me "feel" like there is a rule when dividing radicals with the same radicand but different index' with different index'. Is that true? In this case, how to divide $\frac{\sqrt[3]{64}}{\sqrt{64}}$? I know that the 3rd root and sq roots are 4 and 8 which would leave me with 1/2. using a calculator I can see that the 4th root of $256$ is 4 but I think I'm to arrive at the solution without a calculator.

Is there a prescribed approach or order of operations to simplifying an expression like this?

How can I arrive at $\dfrac{\sqrt{3}}{3}$

Answer 1

Note that $$\sqrt[3]{64}=(64)^{\frac13}=(4^3)^\frac13=4$$ $$\sqrt[4]{256}=(256)^{\frac14}=(4^4)^{\frac14}=4$$ $$\sqrt{64}=8$$ $$\sqrt{256}=16$$ So, we get $$\sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}=\sqrt{\dfrac{4+4}{8+16}}=\sqrt{\dfrac{8}{24}}=\sqrt{\dfrac{1}{3}}=\dfrac{\sqrt{3}}{3}$$

Answer 2

You have: $$\sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$$ Then: $$\sqrt{\frac{\sqrt[3]{8^2} + \sqrt[4]{16^2}}{\sqrt{8^2}+\sqrt{16^2}}}$$ $$\sqrt{\frac{\sqrt[3]{2^6} + \sqrt[4]{4^4}}{8 + 16}}$$ $$\sqrt{\frac{2^2 + 4}{24}}$$ $$\sqrt{\frac{8}{8 \cdot 3}}$$ $$\frac{1}{\sqrt{3}}$$ $$\frac{\sqrt{3}}{\sqrt{3}\sqrt{3}}$$ $$\frac{\sqrt{3}}{3}$$ And this is the answer.

Answer 3

We know that,

$64=8×8$ or $\sqrt{64}=8$

and
$256=16×16$ or $\sqrt{256}=16$

Also, $64^\frac{1}{3}$ and $256^\frac{1}{4}$

Therefore, $64^\frac{1}{3}=(8×8)^\frac{1}{3} =(8^\frac{1}{3})×(8^\frac{1}{3})=2×2=4$

Similarly, $256^\frac{1}{4}=(16×16)^\frac{1}{4}=(16^\frac{1}{4})×(16^\frac{1}{4})=2×2=4$

Now, your expression reduces to

=$\sqrt{\frac{4+4}{8+16}}$

=$\sqrt{\frac{8}{24}}$

=$\sqrt{\frac{1}{3}}$

On rationalization,

$=\frac{1}{\sqrt{3}}×\frac{\sqrt{3}} {\sqrt{3}}$

Hence, $\frac{\sqrt{3}}{3}$

Thanks.

Answer 4

$ 256 $ is a perfect square because $ 16 \cdot 16 = 16^2 = 256 $. Thus $ \sqrt{256} = 16 $. The expression $$ \sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}} $$ evaluates to $$ \sqrt{\frac{4 + 4}{8 + 16}} $$ which simplifies to $$ \sqrt{\frac{8}{24}} $$ Reduce the fraction inside the radical by dividing both the numerator and denominator by the greatest common divisor that divides both $ 8 $ and $ 24 $, which is $ 8 $. $$ \sqrt{\frac{8/8}{24/8}} = \sqrt{\frac{1}{3}} $$

Now from one of the basic properties involving radicals, $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $. Using this property,

$$ \sqrt{\frac{1}{3}} = \frac{\sqrt{1}}{\sqrt{3}} = \frac{1}{\sqrt{3}} $$

"Rationalize" the denominator by multiplying the numerator and denominator by $ \sqrt{3} $ to achieve the desired result.

$$ \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3} $$