What you need to know

Things to remember:

  • If we put a power on something that already has a power, we just multiply them!

“Exponents” is just another way of saying “powers” and tells us how many times we have to multiply a number or letter by itself.

3^4=3\times3\times3\times3

5^6=5\times5 \times5 \times5 \times5 \times5

a^2=a\times a

n^5=n\times n\times n\times n\times n

Simplify (a^5)^3 by writing as a single power.

So, what do we know about these powers?

a^5= a\times a\times a\times a\times a

But, what about the cube power? Well, we do it in the same way, but instead multiply a letter or number by itself 3 times, we are multiply a^5 by itself 3 times.

(a^5)^3=a^5\times a^5 \times a^5

But, we know how to deal with powers when we are multiplying them… we just add!

a^5\times a^5 \times a^5=a^{5+5+5}

Remember though, if we are adding the same number multiple times, we can write it as a multiplication with how many numbers there are!

a^{5+5+5}=a^{5\times3}

And now, all we have to do is multiply these numbers!

a^{5\times3}=a^{15}

So, our answer is:

(a^5)^3 = a^{15}

The important stage here is the one in bold, it shows us that all we have to do is multiply the powers together!

(3^2)^6 = 3^{2\times6}=3^{12}

(6^2)^7 = 6^{2\times7}=6^{14}

(x^5)^9 = x^{5\times9}=x^{45}

(n^3)^y = n^{3\times y}=n^{3y}

(a^b)^c = a^{b\times c}=a^{bc}

Example Questions

Question 1: Simplify (5^7)^3 by writing as a single power.

Answer

(5^7)^3=5^{7\times3}=5^{21}

Question 2: Simplify (a^8)^x by writing as a single power.

Answer

(a^8)^x=a^{8\times x}=a^{8x}