Written as bn, involving two numbers, the bottom b, and therefore the exponent or power n, and pronounced as "b raised to the facility of n".
Written as bn, involving two numbers, the bottom b, and therefore the exponent or power n, and pronounced as "b raised to the facility of n". When n may be a positive integer, exponentiation corresponds to repeated multiplication of the base: that’s, bn is that the product of multiplying n bases:
bn = b x ….. x b/ n times.
The exponent is typically shown as a superscript to the proper of the bottom. therein case, bn is named "braised to the nth power", "braised to the facility of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth".
One has b1 = b, and, for any positive integers m and n, one has bn ⋅ bm = bn+m. to increase this property to non-positive integer exponents, b0 is defined to be 1, and b−n (with n a positive integer and b not zero) is defined as 1/bn. especially, b−1 is adequate to 1/b, the reciprocal of b.
The fractional exponents are how of expressing powers also as roots in one notation.
What does it mean, exactly? Let's have a glance at a couple of simple examples first, where our numerator is adequate to 1:
64 (1/2) = √64
27 (1/3) = ³√27
From the equations above we will deduce that:
An exponent of 1/2 maybe a root
An exponent of 1/3 maybe a root
An exponent of 1/4 maybe the fourth root
An exponent of 1/k may be a kth root
Formula: exponent of 1/k adequate to k-th root
But why is it so? We'll attempt to prove it:
Let's use the law of exponents which says that we will add the exponents when multiplying two powers
Those have an equivalent base:
xa+b = xa * xb
So, for instance, if n = 2
X² = x¹âº¹ = x¹ * x¹ = x * x
Try this with any number you wish, it is often true!
Then, let's check out the fractional exponents of x:
x = x¹ = x (1/2 + 1/2) = x (1/2) * x (1/2)
How does one call variety which - when multiplied by itself - gives another number? It is a root, of course! So we acknowledged that:
X (1/2) = √x
If you wish, you'll analogically check other roots, e.g. the cube root:
x = x (1/3 + 1/3 + 1/3) = x (1/3) * x (1/3) * x (1/3) = ³√x * ³√x * ³√x
So x (1/3) = ³√x
Now we all know that x to the facility of 1 third is adequate to the root of x.
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As we know in fraction exponent we have three value,
Now, based on this fraction exponent we have provided 3 boxes.
Now in their boxes, you have filled the proper value
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