Simpler or shorter way to simplify expression $(16^{2} \times 64^{3})\div1024^{2}$ for a power

Question

I am studying about powers for a discipline in college and the teacher asked me to simplify the following expression to transform it into the form of a single power,

$$ (16^{2} \times 64^{3})\div1024^{2} $$

I can simplify to,

$$ 2^{6} $$

But, take many steps to get this result,

$$ (16^{2} \times 64^{3})\div1024^{2} \\ \implies(16\times16)\times(64\times64\times64)\div(1024\times1024) \\ \implies 256 \times262144\div1048576\\ \implies67108864\div1048576=64\\ 64\implies2^{6} \\ (16^{2} \times 64^{3})\div1024^{2} \implies 2^{6} $$

However I would like to know if there is a shorter or simpler way to simplify expression $(16^{2} \times 64^{3})\div1024^{2}$ ?

Answer 1

\begin{align} & (16^2 \times 64^3)\div1024^2 \\[10pt] = {} & (2^4)^2 \times (2^6)^3 \div (2^{10})^2 \\[10pt] = {} & 2^8 \times 2^{18} \div 2^{20} \\[10pt] = {} & 2^{8+18-20}. \end{align}

Answer 2

Too complicated. Notice that:

• $16 = 2^4$;
• $64 = 2^6$;
• $1024 = 2^{10}$.

Therefore:

$$\begin{array}[rcl] ((16^{2} \times 64^{3})\div 1024^{2} & = & (2^{8} \times 2^{18})\div 2^{20} \\ & = & 2^{26}\div 2^{20} = 2^6 = 64. \\ \end{array}$$

Answer 3

By writing these out as powers of primes (namely $2$), we have $$ \frac{16^{2} \times 64^{3}}{1024^{2}} = \frac{(2^{4})^{2} \times (2^{6})^{3}}{(2^{10})^{2}} = \frac{2^{8}\times2^{18}}{2^{20}} = \frac{2^{26}}{2^{20}} = 2^{6}. $$

Answer 4

Notice that all can be represented in powers of 2

$$16^2=(2^4)^2=2^8$$ $$64^3=(2^6)^3=2^{18}$$ $$1024^2=(2^{10})^2=2^{20}$$ $$\frac{2^8\cdot 2^{18}}{2^{20}}=\frac{2^{26}}{2^{20}}=2^6$$

Answer 5

$$(16^{2} \times 64^{3})\div1024^{2}=2^8 \times 2^{18} /2^{20} =2^6=64$$