** Question Form_labelMessages **
Roots
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If we have the expression x^2=5 to find x we take the square root of 5.
In an expression such as x^5=10 to solve for x we must take the 5th root of ten, which looks like the 10 in a square root bracket with a 5 outside of it.
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I understand to an extent the relation between powers and roots but it is a subject I am mostly unfamiliar with. How do we determine the 5th root of 10 in such an expression as the one above?
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The square root is inverse to the square, in the sense that if you take the square root of a number, then square it, you get the number back. So, as you're aware, the square root of 25 is 5, because squaring 5 gives you 25.
We also have the law of exponents which says that
(a^b)^c = a^(b c).
It's easy to understand this when b and c are integers. For example, (2^3)^4 = (2^3) * (2^3) * (2^3) * (2^3) = (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) = 2^12.
As another example, relevant to the present situation, we let b = 2 and c = 1/2, so that we take the 1/2 power of the second power of a:
(a^2)^(1/2) = a^(2 * 1/2) = a^1 = a.
In general,
(a^n)^(1/n) = a^(n * 1/n) = a^1 = a.
If the square of the square root of a number is equal to the number, then we have to say that
(sqrt(x))^2 = x.
If we take the 1/2 power of both sides of this equation we get
((sqrt(x))^2 )^(1/2) = x^(1/2).
Since the 1/2 power of power 2 is the power 2 * 1/2 = 1, we see that the left-hand side becomes just sqrt(x), so we can write
sqrt(x) = x^(1/2).
We can follow a similar line of reasoning to conclude that the nth root of a number is the 1/n power of that number.
Now, to solve x^5 = 10, we could think in terms of the 5th root (and it would be good to do so), but we can also just think in terms of the 1/5 power.
Given
x^5 = 10
we would raise both sides to the 1/5 power to get
(x^5)^(1/5) = 10^(1/5).
The left-hand side is the 1/5 power of the 5th power; since 1/5 * 5 = 1 the left-hand side is x^1 or just x. Thus our solution is
x = 10^(1/5).
You can use your calculator to evaluate 10^(1/5).
Since 10^(1/5) is the 5th root of 10, we have solved the equation by effectively calculating the 5th root of 10. However, we wrote our solution equivalently in terms of the reciprocal power rather than the root.
It's important to understand both reciprocal power and roots. However since there is not really good, compact way to express radicals on the keyboard, we will use the fractional notation in this course.
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