Question Video: Simplifying an Expression Involving Exponents Mathematics • Second Year of Secondary School

Video Transcript

Simplify negative 64𝑎 to the power of 12 𝑏 to the power of 18 all to the power of one-sixth, where 𝑎 and 𝑏 are positive constants.

When it comes to simplifying a problem like this, we can really consider it as all of the separate parts of this expression taken to the power of one-sixth. So we’ll have negative 64 to the power of one-sixth, 𝑎 to the power of 12 to the power of one-sixth, and 𝑏 to the power of 18 to the power of one-sixth all multiplied together. A useful exponent rule to remember here is that 𝑎 to the power of 𝑛 taken to the power of 𝑚 is equal to 𝑎 to the power of 𝑛𝑚.

The first calculation that we need to do here is 64 to the power of one-sixth. Remember, that’s the same as finding the sixth root of 64. We can work out that two to the power of six is 64. And so we can work out that the sixth root of 64 or indeed 64 to the power of one-sixth is also two. And we mustn’t forget to carry the negative down from the line above. We can then apply our exponent rule to the next part of this calculation 𝑎 to the power of 12 to the power of one-sixth.

Since the exponents of 12 and one-sixth are multiplied, we have 12 times one-sixth, which leaves us with 𝑎 to the power of two or 𝑎 squared. We then have the final part, 𝑏 to the power of 18 to the power of one-sixth. Multiplying the exponents of 18 and one-sixth leaves us with an exponent of three. We can then further simplify the answer as negative two 𝑎 squared 𝑏 cubed.