Changing the base of a power

We know from the page on raising an index to an index that we can rewrite (a^x)^y as a^{xy}.

Example: write 2^6 as a power of 2^2

• =2^{2\times3}
• =(2^2)^3

Example: write 81^3 as a power of 3

We can rewrite 81 as 3^4, so:

• 81^3
• =(3^4)^3
• =3^{4\times3}
• =3^{12}

Example: write 16^4 as a power of 2

We can rewrite 16 as 2^4, so:

• 16^4
• =(2^4)^4
• =2^{4\times4}
• =2^{16}

Example: write 27^5 as a power of 3

We can rewrite 27 as 3^3, so:

• 27^5
• =(3^3)^5
• =3^{3\times5}
• =3^{15}

Example: write 64^2 as a power of 4

We can rewrite 64 as 4^3, so:

• 64^2
• =(4^3)^2
• =4^{3\times2}
• =4^{6}

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