- #1
Albert1
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if $2^w+2^x+2^y+2^z=20.625$
here $w>x>y>z$
and $w,x,y,z \in Z$
find $w+x+y+z$
here $w>x>y>z$
and $w,x,y,z \in Z$
find $w+x+y+z$
Albert said:if $2^w+2^x+2^y+2^z=20.625$
here $w>x>y>z$
and $w,x,y,z \in Z$
find $w+x+y+z$
nice solution !Pranav said:$$20.625=16+4+0.5+0.125$$
$$\Rightarrow 20.625=2^4+2^2+2^{-1}+2^{-3}$$
$$w+x+y+z=4+2-1-3=\boxed{2}$$
To find the values of $w,x,y,z$ in this equation, we can use logarithms. Specifically, we can take the logarithm of both sides of the equation to get:
$log(2^w+2^x+2^y+2^z)=log(20.625)$
Using logarithmic properties, we can rewrite the left side as:
$log(2^w)+log(2^x)+log(2^y)+log(2^z)=log(20.625)$
Then, we can use the power rule of logarithms to simplify the equation to:
$wlog(2)+xlog(2)+ylog(2)+zlog(2)=log(20.625)$
Finally, we can plug in the value of $log(2)$, which is approximately 0.301, and solve for each variable.
Yes, the decimal in the sum is a result of the logarithmic calculation. When we take the logarithm of a number, we are essentially finding the exponent that the base needs to be raised to in order to get that number. Since 20.625 is not a power of 2, the sum of the exponents (which represent $w,x,y,z$) will result in a decimal value.
Yes, in this particular equation, we can use the fact that 20.625 is equal to $2^4+2^{0.5}$. This means that $w=4$ and $x=0.5$. Then, we can use basic algebra to solve for $y$ and $z$. This method is faster because we do not need to use logarithms and can immediately see the values of two variables.
No, $w,x,y,z$ cannot be negative numbers in this equation. Since negative exponents represent fractional values, they would result in a decimal sum that is less than 20.625. However, the equation must have a sum of exactly 20.625, so the exponents must be positive integers.
Yes, there are infinitely many solutions to this equation. In fact, for any given value of $w$, we can find values of $x,y,z$ that satisfy the equation. For example, if $w=2$, then $x=y=z=0.5$ also satisfies the equation. This is because $2^2+2^{0.5}+2^{0.5}+2^{0.5}=20.625$. However, using logarithms is the most efficient method to find a specific solution to the equation.