Calculate negative four-fifths to the third power times negative four-fifths squared.
Let’s start this question by recalling that a number squared, for example, 𝑥 squared, is the same as 𝑥 times 𝑥. A number written to the third power is equivalent to that number times itself three times. We could say that 𝑥 to the third power is equal to 𝑥 times 𝑥 times 𝑥. So if we take the first term in our calculation, negative four-fifths to the third power, then this would be equal to negative four-fifths times negative four-fifths times negative four-fifths. And it’s very important to notice that the negative sign is part of the number that’s multiplied by itself three times. Now, we will look at the second part of our calculation, the negative four-fifths squared. And this is equivalent to negative four-fifths times negative four-fifths. And since our fraction to the third power is multiplied by our fraction squared, then we do the equivalent in the expanded version.
So now we have the fraction negative four over five multiplied by itself five times. Then we can write this in a neater way, as negative four-fifths to the fifth power. And that’s because we know that a number to the fifth power is equal to that number multiplied by itself five times. If we look again at the question, we can see that there is another rule of exponents that we could’ve used to help us answer this. And that is if we have 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏, that equals 𝑥 to the power of 𝑎 plus 𝑏. In this case, our value of 𝑥 would be equivalent to negative four-fifths. Our first exponent 𝑎 is equivalent to three. And our second exponent 𝑏 will be equivalent to two. So our answer 𝑥 to the power of 𝑎 plus 𝑏 would be minus four-fifths to the power of five, since five is equal to three plus two.
But using either method would give us the final answer of negative four-fifths to the power of five.