Video Transcript
Simplify 0.8 raised to the power of one over four multiplied by 36 raised to the power of one over eight multiplied by five to the power of three over four all divided by 30 to the power of negative seven over four times 1.25 raised to the power of one-quarter.
In this question, we’re asked to simplify an expression involving the product and quotient of exponential expressions. This means we’re going to need to simplify this expression by using the laws of exponents. However, we can see that this is a very complicated expression. So instead, let’s simplify each factor in the numerator and denominator separately. We can then use this to simplify the entire expression at the end. Let’s start with the first factor in the numerator. That’s 0.8 raised to the power of one-quarter.
When simplifying expressions of this form, it’s usually a good idea to write the base as a fraction. And this gives us four over five all raised to the power of one-quarter. And then after this, we should factor both the numerator and denominator of our fraction into primes. And the reason for this is the laws of exponents usually require either the same base or the same exponent when simplifying. Therefore, the easiest way to simplify an expression involving many different bases is to try and make the bases as simple as possible. We do this by factoring them into primes. Since four is equal to two squared and five is a prime, this gives us two squared over five all raised to the power of one-quarter.
And now we can simplify this further by using the laws of exponents. We recall 𝑎 over 𝑏 all raised to the 𝑛th power is equal to 𝑎 to the 𝑛th power divided by 𝑏 to the 𝑛th of power. In this case, our value of 𝑎 is two squared, our value of 𝑏 is five, and 𝑛 is one-quarter. This gives us two squared all raised to the power of one-quarter divided by five raised to the power of one-quarter.
And finally, we can simplify the numerator of this expression by noticing we’re raising two to the power of two and then raising this all to the power of one-quarter. And when we raise a base to an exponent and raise all of this to an exponent, we multiply the exponents. 𝑎 to the power of 𝑛 all raised to the power of 𝑚 is 𝑎 to the power of 𝑛 times 𝑚. And two times one-quarter is equal to one-half. So this gives us two to the power of one-half divided by five to the power of one-quarter. And we could rewrite this by using radicals. However, when we’re simplifying exponential expressions, it’s usually a good idea to leave the exponents as a number. This will make simplification easier to perform later, since we may need to perform arithmetic on the exponents.
Now, we can move on to the second factor in our numerator. To do this, we’ll start by clearing some space. We need to simplify 36 raised to the power of one-eighth. We’ll do this by using a similar process. First, we’re going to factor the base into primes. 36 is two squared multiplied by three squared. This gives us two squared times three squared all raised to the power of one-eighth. And now that our base is a product, we can distribute the exponent over this product. 𝑎 times 𝑏 all raised to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 times 𝑏 to the power of 𝑛. This gives us two squared raised to the power of one over eight multiplied by three squared raised to the power of one over eight.
And finally, since each factor is a base raised to an exponent all raised to another exponent, we can simplify each factor by multiplying the exponents. In both factors, we get a new exponent of two times one over eight, which is one-quarter. So this simplifies to give us two to the power of one-quarter multiplied by three to the power of one-quarter. Now, we would move on to try and simplify the third factor in our numerator. However, the base is already a prime number and the exponent is already rational. So this factor is already simplified.
Let’s now move on to the first factor in the denominator. We could clear some space. However, this simplification process is going to follow the exact same process it did for 36 to the power of one over eight. First, we factor the base into primes. We see that 30 is equal to two times three times five. Then, we use our laws of exponents to distribute the exponent over this product. This gives us two to the power of negative seven over four multiplied by three to the power of negative seven over four multiplied by five to the power of negative seven over four. And this is now a product of bases to rational exponents where all of the bases are prime numbers. So we don’t need to simplify this any further.
We can in fact follow a very similar process to simplify the final factor in the denominator. Once again, we’ll start by writing the base as a fraction and then factoring the numerator and denominator into primes. This gives us five divided by two squared all raised to the power of one-quarter. Now, we want to distribute the exponent over our parentheses. And to do this, we recall 𝑎 over 𝑏 all raised to the 𝑛th power is equal to 𝑎 to the 𝑛th power divided by 𝑏 to the 𝑛th power. This then gives us five to the power of one-quarter divided by two squared all raised to the power of one-quarter. And in our denominator, we have two raised to an exponent raised to another exponent.
Remember, this will be the same as two raised to the power of the product of the exponents: two to the power of two times one-quarter, which is two to the power of one-half. Therefore, our final factor simplifies to give us five to the power of one-quarter divided by two to the power of one-half.
Now that we’ve simplified all of these expressions, let’s clear some space so we can substitute these into our expression. We want to substitute all of these expressions at once; however, this will give us a very complicated expression. Instead, we notice 0.8 to the power of one-quarter is in our numerator. So when we substitute in the expression two to the power of one-half all divided by five to the power of one-quarter, we can instead add a factor of two to the power of one-half into the numerator and the factor of five to the power of one-quarter in the denominator. And this is just because dividing the numerator by five to the power of one-quarter is the same as multiplying the denominator by five to the power of one-quarter.
We can do something similar for 1.25 raised to the power of one-quarter in our denominator. Instead of just substituting five to the power of one-quarter divided by two to the power of one-half, we can instead add a factor of five to the power of one-quarter in the denominator and a factor of two to the power of one-half in the numerator. This is because dividing the denominator by two to the power of one-half will be the same as multiplying the numerator by two to the power of one-half. Therefore, if we substitute these expressions in and rearrange slightly, we get the following expression. And this is now the product and quotient of exponential expressions with prime bases and rational exponents. And this will now allow us to simplify this expression by using our laws of exponents.
We could simplify this expression all at once; however, we’ll simplify the numerator and denominator separately. First, we recall if we multiply two exponential expressions with the same base, we add the exponents. 𝑎 to the power of 𝑛 times 𝑎 to the power of 𝑚 is 𝑎 to the power of 𝑛 plus 𝑚. We can use this to simplify our numerator. We have two to the power of one-half times two to the power of one-quarter times two to the power of one-half. We need to add these exponents together. That’s one-half plus one-quarter plus one-half. And if we evaluate one-half plus one-quarter plus one-half, it’s five over four. So this factor simplifies to give us two to the power of five over four.
We can then use this rule to simplify the factors with base three or base five in our numerator since these bases only appear once as factors. So we’ll leave these factors as they are for now in the numerator. And we get a very similar story in the denominator. The bases of two and three only appear as single factors. So we’ll leave these two factors as they are for now. We can use this rule to simplify all of the factors in the denominator with base five. We use the same rule. Since the bases of these three numbers are all the same, we need to add their exponents. That’s one-quarter plus negative seven over four plus one-quarter, which if we calculate is equal to negative five over four. Therefore, we can simplify this factor in our denominator to five to the power of negative five over four. This gives us the following expression.
And now we can see we’re taking the quotient of exponential expressions with the same base. And we can simplify this by using our laws of exponents. 𝑎 to the power of 𝑛 divided by 𝑎 to the power of 𝑚 is 𝑎 to the power of 𝑛 minus 𝑚. We find the difference of the exponents. And it’s worth pointing out we could’ve applied both of these steps at the same time. For example, to find the factor of two, we will need to add all the exponents of two in the numerator and then subtract all of the exponents of two in the denominator. However, it’s personal preference which one you would want to use.
Let’s start by using this to simplify two to the power of five over four divided by two to the power of negative seven over four. We need to subtract the exponents. We get five-quarters minus negative seven over four, which if we calculate is equal to 12 over four, which is three. Therefore, we get a factor of two cubed. We can do the same with three to the power of one-quarter divided by three to the power of negative seven over four. We find the difference in the exponents. It’s eight divided by four, which is equal to two. And this is the exponent of three, so we get a factor of three squared.
Finally, we’ll do the same with our base of five. Three-quarters minus negative five-quarters is eight-quarters, which is equal to two. Therefore, we have a factor of five squared in our answer. It’s two cubed times three squared times five squared. And finally, we can evaluate this expression. We recall two cubed is two times two times two, three squared is three times three, and five squared is five times five, which if we calculate is 1,800, which is our final answer.
