# How do you simplify (5^8)^3?

Jan 10, 2016

Explanation is given below.

#### Explanation:

Exponent rules or sometimes called as laws of exponents.
You can go over the rules and would be in a position to solve many such problems as you have shared.

Let us understand exponents in easier manner.

${a}^{1} = a$
${a}^{2} = a \cdot a$
${a}^{3} = a \cdot a \cdot a$

You can see the exponent denotes the number of time the base is to be multiplied with itself.

A very common rule is

${a}^{m} \cdot {a}^{n} = {a}^{m + n}$

You should be familiar with it, if not take some time and go over the rules it would be such a life saver later on in Maths.

Now let us come problem similar to ours.

Example : ${\left({a}^{2}\right)}^{3}$

This we can write as ${a}^{2} \cdot {a}^{2} \cdot {a}^{2}$ that is what we understand by exponents.

Now applying the rule we can see

${\left({a}^{2}\right)}^{3} = {a}^{2} \cdot {a}^{2} \cdot {a}^{2}$
${\left({a}^{2}\right)}^{3} = {a}^{2 + 2 + 2}$
${\left({a}^{2}\right)}^{3} = {a}^{6}$

Now I would like to point out that the product of the two exponents here that is $2$ and $3$ also gives us $6$

To generalize

${\left({a}^{m}\right)}^{n} = {a}^{m \times n}$

The above rule can be used for our problem.

${\left({5}^{8}\right)}^{3}$

$= {5}^{8 \times 3}$

$= {5}^{24}$ Answer.

Jan 10, 2016

The answer would be ${5}^{24}$, as ${\left({x}^{m}\right)}^{n} = {x}^{m n}$

#### Explanation:

We understand that raising something to a power produces a product of that number and itself as many times as the number the power is.
e.g. ${6}^{3} = 6 \cdot 6 \cdot 6$
So ${\left({5}^{8}\right)}^{3} = {5}^{8} \cdot {5}^{8} \cdot {5}^{8}$

By our law of indicies: ${x}^{n} \cdot {x}^{m} = {x}^{n + m}$

We therefore have: ${\left({5}^{8}\right)}^{3} = {5}^{8} \cdot {5}^{8} \cdot {5}^{8} = {5}^{8 + 8 + 8} = {5}^{24}$

Taking it even simpler, ${5}^{8} = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

So: ${5}^{8} \cdot {5}^{8} \cdot {5}^{8} = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = {5}^{24}$

Overall, the rule is, if you have a number raised to a power, and both the number and power are raised to another power, you can simply multiply the two powers together to get the new power for the number.

e.g: ${\left({x}^{m}\right)}^{n} = {x}^{m n}$