Math CentralQuandaries & Queries


Question from Shalaine, a student:

What is the exponential form of (7cubed * 8(exponent6))exponent 6?

Hi Shalaine,

There is no such thing as "the exponential form." Exponential form just means involving exponents Thus

\[(7^3 \times 8^6)^6 \]

is already in exponential form. I expect that the problem you were given is to express this expression a more compact form.

If you express $7^3 \times 8^6$ in a non-exponential form it is the product of three sevens times the product of six eights. This is then raised to the sixth power so it is repeated six times. So now you have $6 \times 3 = 18$ sevens and $6 \times 6 = 36$ eights. Thus the final result is $7^{18}\times 8^{36}.$

For a more traditional, mathematical solution you can use one of the laws of exponents which says

\[(a^b)^c = a^{b \times c}.\]


\[(7^3 \times 8^6)^6 = (7^3)^6 \times (8^6)^6 = 7^{3 \times 6} \times 8^{6 \times 6} = 7^{18} \times 8^{36.}\]


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