Evaluate expression without using a calculator

[Nazariy]

Hello,

I am new here, so I might have posted this in a wrong place, if so, excuse me.

I have got a very trivial question, how do I solve 16^-3/4 without a calculator. I know it is 1/16^3/4 but that is it, I do not know how to evaluate it further. Thank you for your help.

 

[srmichael]

Hello,

I am new here, so I might have posted this in a wrong place, if so, excuse me.

I have got a very trivial question, how do I solve 16^-3/4 without a calculator. I know it is 1/16^3/4 but that is it, I do not know how to evaluate it further. Thank you for your help.

Do you know that a^(m/n) = [a^(1/n)]^m

Use this little diddy for 16^(3/4)

 

[Quaid]

Hi Nazariy:

It also helps to know that 16^(1/4) is exponential notation for "the fourth root of 16".

Cheers :cool:

 

[Nazariy]

Thank you for the tips.

So it will basically be one divided by three fourth roots of 16. I would still need to know what a fourth root of 16 equals to. Square root would be four and thus a fourth root should be 2 or am I missing something? If so, then 16^-3/4 should be 1/(2*2*2), thus 1/8th.

Yeap, and that is correct. I however, intuitively guessed that the fourth root of 16 is 2, I will have to look into this more. Thank you,

 

[Nazariy]

Oh I think I know, a fourth root is a square root of a square root!

thanks!

 

[Dale10101]

Nice

Thank you for the tips.

So it will basically be one divided by three fourth roots of 16. I would still need to know what a fourth root of 16 equals to. Square root would be four and thus a fourth root should be 2 or am I missing something? If so, then 16^-3/4 should be 1/(2*2*2), thus 1/8th.

Yeap, and that is correct. I however, intuitively guessed that the fourth root of 16 is 2, I will have to look into this more. Thank you,

\[{16^{\frac{{ - 3}}{4}}} = \frac{1}{{{{16}^{\frac{3}{4}}}}} = \left( {\frac{1}{{{{16}^{\frac{1}{4}}}}}} \right)\left( {\frac{1}{{{{16}^{\frac{1}{4}}}}}} \right)\left( {\frac{1}{{{{16}^{\frac{1}{4}}}}}} \right) = \left( {\frac{1}{{\sqrt[4]{{16}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{16}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{16}}}}} \right) = \left( {\frac{1}{{\sqrt[4]{{{2^4}}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{{2^4}}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{{2^4}}}}}} \right) = \left( {\frac{1}{2}} \right)\left( {\frac{1}{2}} \right)\left( {\frac{1}{2}} \right) = \frac{1}{8}\]

Just as you calculated.

To find the 4th root of 16 you ask yourself the question, "Four factors of what number multiply to make 16", i.e. (n)(n)(n)(n) = 16, n = ?

Square root of the square root is good too: \[{16^{\frac{1}{4}}} = {\left( {{{16}^{\frac{1}{2}}}} \right)^{\frac{1}{2}}} = {\left( {\sqrt {16} } \right)^{\frac{1}{2}}} = \sqrt {\sqrt {16} } \]

 

[Nazariy]

\[{16^{\frac{{ - 3}}{4}}} = \frac{1}{{{{16}^{\frac{3}{4}}}}} = \left( {\frac{1}{{{{16}^{\frac{1}{4}}}}}} \right)\left( {\frac{1}{{{{16}^{\frac{1}{4}}}}}} \right)\left( {\frac{1}{{{{16}^{\frac{1}{4}}}}}} \right) = \left( {\frac{1}{{\sqrt[4]{{16}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{16}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{16}}}}} \right) = \left( {\frac{1}{{\sqrt[4]{{{2^4}}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{{2^4}}}}}} \right)\left( {\frac{1}{{\sqrt[4]{{{2^4}}}}}} \right) = \left( {\frac{1}{2}} \right)\left( {\frac{1}{2}} \right)\left( {\frac{1}{2}} \right) = \frac{1}{8}\]

Just as you calculated.

To find the 4th root of 16 you ask yourself the question, what multiplied by itself 4 times equal 16.

Square root of the square root is good too: \[{16^{\frac{1}{4}}} = {\left( {{{16}^{\frac{1}{2}}}} \right)^{\frac{1}{2}}} = {\left( {\sqrt {16} } \right)^{\frac{1}{2}}} = \sqrt {\sqrt {16} } \]

Perfect, thank you!

Off-topic question, how do I insert formulas as well?

 

[mmm4444bot]

how do I insert formulas

If you're asking about the mathematical typesetting in this thread, it's called LaTex. (Right-click on any LaTex formatting, and use the menus to explore the coding.)

You may google LaTex for information, including tutorials.

Be aware that there are many different versions of LaTex around the Internet; not everything out there works on these boards. For example, we do not invoke LaTex using $ symbols. Enclose your coding between [ֹtex] and [/ֹtex] tags, instead.

Cheers :cool:

 

[Nazariy]

If you're asking about the mathematical typesetting in this thread, it's called LaTex. (Right-click on any LaTex formatting, and use the menus to explore the coding.)

You may google LaTex for information, including tutorials.

Be aware that there are many different versions of LaTex around the Internet; not everything out there works on these boards. For example, we do not invoke LaTex using $ symbols. Enclose your coding between [ֹtex] and [/ֹtex] tags, instead.

Cheers :cool:

Awesome, thank you!