Question

How do you determine which is greater `4^2*4^3 (<=>) 4^5`?

Answer 1

`color(green)(4^2*4^3 = 4^5`

Explanation:

One of the properties of exponents says that
`color(blue)(a^m * a^n = a^(m+n)`

Hence `4^2 * 4^3 = 4^(2+3) = 4^5`

Hence we can say that `color(green)(4^2*4^3 = 4^5`

Answer 2

They are equal.

Explanation:

The rule says that the multiplication of powers of a same base is the base elevated to the sum of the powers. In formulas: `a^b * a^c = a^{b+c}`. The reason is very simple: let's prove that `4^2*4^3=4^5`.

Apply the definition of power: `4^2=4*4`, and `4^3=4*4*4`.

Multiply the two numbers:

`(4*4)*(4*4*4) = 4*4*4*4*4=4^5`

So, no one of the two numbers is greater than the other, they are exactly the same.

Answer 3

Based on an assumption: They are the same!

Explanation:

Assumption: by the dot you mean multiply.

Consider `2 times 2`. This may be written as `2^2` ..(1)

Now consider `2 times 2 times 2 times 2`. This may be written as `2^4` ..(2)

Now consider both parts (1) and (2) together:

(2) may be rewritten as `2^2 times 2^2`
but we know that (2) is also `2^4` so:

`2^2 times 2^2 = 2^4`

From this we can deduce that `2^2 times 2^2 = 2^(2+2)`

Ok! now look at the question given:

We have `4^2 times 4^3`

By using the same approach as before this would give us:

`4^(2+3) = 4^5`

In conclusion: both sides of `(<=>)` have the same value so neither is greater.