Difference between revisions of "1959 AHSME Problems/Problem 5"

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==Solution==
 
==Solution==
When we multiply numbers with exponents, we add tne exponents together and leave the bases unchanged. We can apply this concept to computate <math>256^{0.16} \cdot 256^{0.09}</math>:  
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When we multiply numbers with exponents, we add the exponents together and leave the bases unchanged. We can apply this concept to computate <math>256^{0.16} \cdot 256^{0.09}</math>:  
 
<cmath>256^{0.16} \cdot 256^{0.09} = 256^{0.16+0.09}=256^{0.25}.</cmath>
 
<cmath>256^{0.16} \cdot 256^{0.09} = 256^{0.16+0.09}=256^{0.25}.</cmath>
 
Now we can convert the decimal exponent to a fraction: <cmath>256^{0.25} = 256^{\frac{1}{4}}.</cmath>
 
Now we can convert the decimal exponent to a fraction: <cmath>256^{0.25} = 256^{\frac{1}{4}}.</cmath>

Latest revision as of 22:08, 25 February 2018

Problem 5

The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is:

$\textbf{(A)}\ 4 \qquad \\ \textbf{(B)}\ 16\qquad \\ \textbf{(C)}\ 64\qquad \\ \textbf{(D)}\ 256.25\qquad \\ \textbf{(E)}\ -16$

Solution

When we multiply numbers with exponents, we add the exponents together and leave the bases unchanged. We can apply this concept to computate $256^{0.16} \cdot 256^{0.09}$: \[256^{0.16} \cdot 256^{0.09} = 256^{0.16+0.09}=256^{0.25}.\] Now we can convert the decimal exponent to a fraction: \[256^{0.25} = 256^{\frac{1}{4}}.\] Now, let us convert the expression into radical form. Since $4$ is the denominator of the fractional exponent, it will be the index exponent: \[256^{\frac{1}{4}}=\sqrt[4]{256}.\] Since $256 =16^2=(4^2)^2=4^4$, we can solve for the fourth root of $256$: \[\sqrt[4]{256}=\sqrt[4]{4^4}=4.\] Therefore, $(256)^{.16} \cdot (256)^{.09}=\boxed{\textbf{(A) }\ 4}.$

See also

1959 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
All AHSME Problems and Solutions

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