How can I calculate manually (1/8) to the power of (-2/3)? Thanks in advance.
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To calculate 1/8 to the power of -2/3, or `(1/8)^(-2/3)` , you need to know the following rules of exponents:
The fractional exponent m/n can be rewritten as
`a^(n/m) = root(m) (a^n) = (root(m) a)^n` . In other words, the numerator n of the fraction stays the power of base a, but the denominator m becomes the index of the radical, or mth degree root of a^n.
The negative exponent can be rewritten as
`a^(-n) = 1/a^n` . In other words, negative exponent -n is the positive exponent n of the reciprocal of base a.
These rules can be applied in any order. First, we can rewrite the given expression without the negative exponent. It will be positive power 2/3 of the reciprocal of 1/8, which is 8:
`(1/8)^(-2/3) = 8^(2/3)`
Next, rewrite the fractional exponent as a radical. It will be the third degree (cube) root of 8 squared:
`8^(2/3) = root(3) (8^2) = (root(3) 8)^2`
As you can see, you can either square 8 first, and then take cube root of 8 squared, or you can take cube root of 8 first, and then square the result. Usually, it is easier to take the root first.
By inspection, you can see that cube root of 8 is 2: `root(3) 8 = 2` . This is because the third power of 2 is 8: `2^3 = 2*2*2 = 8` .
Finally, square the cube root of 8: `(root(3) 8)^2 = 2^2 = 4` .
The final answer is 4.
When you have something to a negative power, you can "flip" it to get rid of the negative power.
For example `2^-1 = 1/2`
In our case `(1/8)^(-2/3) = 8^(2/3)`
Then we can write this as the third root of 8, squared. The third root coming from 3 being in the denominator, and the squared coming from the 2 in the numerator of the exponent.
`root(3)(8)^2`
The third root of 8 is 2,
`2^2`
And 2 squared is 4!
The final answer is 4.
Hope this helps!
I would suggest breaking this math up into separate steps. Taking a number to the power of -2/3 can be thought of as the following steps: cube root it (the 1/3 part), square it (the 2 part), and take its reciprocal (the – part). In that case, let’s walk through each step individually:
- Cube root it: To take the cube root of a fraction easily by hand, remember that the cube root of the fraction is equal to the cube root of the numerator divided by the cube root of the denominator. This greatly simplifies our math. The cube root of 1 is 1, and the cube root of 8 is 2. Therefore, the result we have from this step is ½.
- Square it: Similar to the idea from when we were taking the cube root, think about squaring the numerator and denominator separately then pulling it back together. Since 1 squared is 1 and 2 squared is 4, at this point we have ¼.
- Taking the reciprocal: Easy enough – all you have to do is flip the numerator and denominator. This gives you 4/1, which can be simplified to just 4. Therefore, you answer for (1/8) to the power of (-2/3) is 4.