Question

How do you simplify and write `(-5.3)^0` with positive exponents?

Answer 1

`(-5.3)^0=1`

Explanation:

Remember the identity `a^m-:a^n=a^(m-n)`, for all `a`, where `m` and `n` are two natural numbers. For example

`(5.3)^3-:(5.3)^2=(5.3xx5.3xx5.3)/(5.3xx5.3)`

= `(cancel(5.3xx5.3)xx5.3)/(cancel(5.3xx5.3))`

= `5.3` and is nothing but `5.3^((2-1))=5.3^1=5.3`

Similarly `(-7.9)^5-:(-7.9)^3`

= `((-7.9)xx(-7.9)xx(-7.9)xx(-7.9)xx(-7.9))/((-7.9)xx(-7.9)xx(-7.9))`

= `((-7.9)xx(-7.9))=(-7.9)^2=(-7.9)^2=(-7.9)^((5-3))`

and hence `(-2.7)^3-:(-2.7)^3=(-2.7)^((3-3))=(-2.7)^0`

or `(-2.7)^3-:(-2.7)^3=((-2.7)xx(-2.7)xx(-2.7))/((-2.7)xx(-2.7)xx(-2.7))=1`

Hence for any number `a`,

if `m=n`, we get `a^m-:a^m=a^(m-m)`

or `a^m/a^m=a^(m-m)`

or `1=a^0`

Hence zero power of any number `a` is `1`

Hence `(-5.3)^0=1`.

Answer 2

In support of Shwetank's answer

Explanation:

Further demonstration of what is happening by example

Suppose we had `3^2/3^4`

Write as `(3xx3)/(3xx3xx3xx3)`

This is the same as: `3/3xx3/3xx1/3xx1/3" "=" "1xx1xx1/3^2`

Another way of writing `1/3^2" "` is `" "color(magenta)(3^(-2))`
'...............................................................................
Lets look at this example again but in another way

write`" "3^2/3^4" as "3^2xx3^(-4)`

This can also be written as `" "3^(2-4)" which is the same as "color(magenta)(3^(-2))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering your question")`

Consider `5^x` where `x` can by any whole number (integer)

Suppose we gave the value of 3 to `x` then we have`" "5^3`

Suppose we had `5^3/5^3` then by the method in the example I gave you we can write this as `" "5^(3-3)=5^0`

But `5^3/5^3=1`

So `1=5^3/5^3=5^(3-3)=5^0 = 1`

`color(magenta)("So a number raised to the power of 0 equals 1")`

`color(green)("In the question the minus is inside the bracket.")`
`color(green)("The index (power) of 0 is applied to everything inside")`
`color(green)("the bracket. So "color(magenta)((-5)^0=1))`
'............................................................................................

Note that `-5^0->-(5^0)=-1`
Test this out on a calculator.