Simplify 0.25 to the power of three over two times 1.8 squared over eight to the power of two-thirds.
In order to help us answer this question, we should recall these five exponent laws. The way that we will approach this question is to take the values of 0.25 and 1.8 and firstly express these as fractions. Once we have done that, we’ll then take those fractions and this value of eight and express these in terms of their prime factorizations.
So let’s begin with 0.25. Well, we know that as a fraction, it’s equal to one-quarter, one over four. When we write the numerator one as a product of its primes, it will simply be one. The value of four is equal to two times two, or two squared. The second part then is 1.8. We can write 1.8 as a fraction nine over five. When we consider the prime factorization then, three squared gives us nine, so that’s our numerator. But the denominator will stay the same as five. And so we can replace 1.8 in the calculation with three squared over five. For the final part then, we’ll just consider the value of eight, and we’ll ignore the exponent for the meantime.
We know that two times two times two gives us eight, so we can write eight as two cubed. And so on the denominator of our entire calculation we’ll have two cubed to the power of two-thirds. At this point, it might look like we’ve actually made our calculation even more complicated. But don’t forget, we have these rules that will help us simplify. Let’s take this one piece at a time. So we’ll use this exponent rule that 𝑎 over 𝑏 to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 over 𝑏 to the power of 𝑛. This means that the first part of our calculation becomes one to the power of three over two over two squared to the power of three over two. And we know that one to the power of three over two is simply one.
We can then simplify this denominator here by using this fourth exponent rule that 𝑎 to the power of 𝑛 to the power of 𝑚 is equal to 𝑎 to the power of 𝑛𝑚. And so multiplying the exponent two and three over two leaves us with three. And so we have one over two cubed. Let’s follow the same process then for three squared over five all squared. And so we’ll have three to the power of two to the power of two over five to the power of two, which simplifies to three to the power of four over five squared. To simplify two cubed to the power of two-thirds, we multiply the exponents, which leaves us with two squared on the denominator.
In order to simplify this and remove the fractions within the fraction, we can take these denominators of two to the power of three and five squared and write them on the denominator of this whole fraction, which gives us one times three to the power of four over two squared times two cubed times five squared. Simplifying this further, the numerator becomes three to the power of four. Then we can use this exponent rule to add the indices on two squared and two cubed to give us two to the power of five multiplied by five squared on the denominator.
We could leave our answer in this simplified form, or, alternatively, we could find the actual numerical values. Three to the power of four is 81. Two to the power of five is 32. And five squared is 25. Working out 32 multiplied by 25 gives us a value of 800. And so we can give the simplified answer to this calculation as 81 over 800.