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

Oct 13, 2015

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 ${4}^{2} \cdot {4}^{3} = {4}^{2 + 3} = {4}^{5}$

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

Oct 13, 2015

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: ${a}^{b} \cdot {a}^{c} = {a}^{b + c}$. The reason is very simple: let's prove that ${4}^{2} \cdot {4}^{3} = {4}^{5}$.

Apply the definition of power: ${4}^{2} = 4 \cdot 4$, and ${4}^{3} = 4 \cdot 4 \cdot 4$.

Multiply the two numbers:

$\left(4 \cdot 4\right) \cdot \left(4 \cdot 4 \cdot 4\right) = 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = {4}^{5}$

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

Oct 13, 2015

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 $\left(\iff\right)$ have the same value so neither is greater.