1959 AHSME Problems/Problem 5
Problem 5
The value of 
is:
Solution
When we multiply numbers with exponents, we add tne exponents together and leave the bases unchanged. We can apply this concept to computate 
:
![\[256^{0.16} \cdot 256^{0.09} = 256^{0.16+0.09}=256^{0.25}.\]](http://latex.artofproblemsolving.com/3/1/3/31384237d46ca82b0154788ab73b9df0bc2c89c5.png)
Now we can convert the decimal exponent to a fraction: ![\[256^{0.25} = 256^{\frac{1}{4}}.\]](http://latex.artofproblemsolving.com/c/1/d/c1d16907bb57ffe5cd7691716308a373307be0c6.png)
Now, let us convert the expression into radical form. Since 
is the denominator of the fractional exponent, it will be the index exponent: ![\[256^{\frac{1}{4}}=\sqrt[4]{256}.\]](http://latex.artofproblemsolving.com/c/1/f/c1f0d3747b9ac71a5fcace8ede61809f935fa386.png)
Since 
, we can solve for the fourth root of 
: ![\[\sqrt[4]{256}=\sqrt[4]{4^4}=4.\]](http://latex.artofproblemsolving.com/6/f/8/6f8b2a71bcd4bb50876530774b5c9972f0190ab9.png)
Therefore, 
See also
| 1959 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
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