Given that 𝑥 equals a half, 𝑦 equals negative two-thirds, and 𝑧 equals negative a third, find the numerical value of 𝑥𝑦 squared over 𝑦 plus 𝑧 squared.
So now if we take a look at the notation and think about how we write algebra, we’ve got 𝑥𝑦 squared as our numerator. And this means 𝑥 multiplied by 𝑦 multiplied by 𝑦, and that’s because 𝑦 multiplied by 𝑦 is 𝑦 squared. Then the denominator is gonna be 𝑦 plus 𝑧 multiplied by 𝑧. That’s because 𝑧 multiplied by 𝑧 is 𝑧 squared.
Okay, so now we understand the notation, how do we start solving this problem? Well, to solve this problem, what we’re gonna do is actually substitute in the values for 𝑥, 𝑦, and 𝑧 that we have. But the best way to do this, the way that I’m gonna do it, is actually to split them up. So do the numerator first and then the denominator and then bring it back together.
So for the numerator, we’re gonna need 𝑥 equals a half and 𝑦 equals negative two-thirds. So we’ve got 𝑥𝑦 squared for our numerator, which is gonna be equal to a half, cause that’s our 𝑥 value, multiplied by then we’ve got negative two over three multiplied by negative two over three, because that was negative two over three squared.
Well, now to actually work this out, we’re gonna use one of our operation rules for fractions. And this is the multiplication rule. We know that if we multiply two fractions, we’re just gonna multiply the numerators and then multiply the denominators. So you have 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑 is equal to 𝑎𝑐 over 𝑏𝑑. And this remains the same irrespective of how many fractions you have.
So this is gonna give us a half multiplied by four over nine. That’s because we had two multiplied by two, which is four, and we have three multiplied by three, which is nine. But both of these are negative, and a negative multiplied by a negative is a positive, so we get four over nine.
And here we actually multiply the numerator and denominator again. So we get one multiplied by four, which is four, and two multiplied by nine, which is 18. And then we can actually simplify because we can divide the numerator and denominator by two, because two is a factor of four and 18. When we do that, we get two-ninths, so we can say that 𝑥𝑦 squared is equal to two-ninths and we’ve dealt with the numerator.
So now we can move on to the denominator. So for the denominator, we have 𝑦 plus 𝑧 squared, which is gonna be negative two over three or negative two-thirds, cause that’s our 𝑦, then plus we’ve got a negative third multiplied by negative a third, because that’s our 𝑧 squared. So then we’re gonna have negative two-thirds plus one-ninth. That’s cause we have one multiplied by one, which gives us one, three multiplied by three, which gives us nine. And it’s a negative multiplied by a negative, which gives us positive. So we got one over nine.
So now what we need to do to actually add our fractions is make sure they have a common denominator. And the common denominator that we can actually have is nine, because three is a factor of nine. So what I’ve done is I’ve converted negative two-thirds to negative six over nine, and I’ve done that by multiplying the numerator by three to go from negative two to negative six and I did that because we multiplied the denominator by three to go from three to nine.
Okay, so we’ve got negative six over nine plus one over nine, which is gonna give us a result of negative five over nine or negative five-ninths. Okay, great! So we’ve actually found the value of the numerator and the value of the denominator. So now let’s bring them back together.
So when we bring them back together, we can have 𝑥𝑦 squared over 𝑦 plus 𝑧 squared. And when we do that, we get two-ninths divided by negative five-ninths, and that’s because two-ninths was the numerator’s value and negative five-ninths was the denominator’s value. And we’ve written it as divided by because actually 𝑥𝑦 squared over 𝑦 plus 𝑧 squared is the same as 𝑥𝑦 squared divided by 𝑦 plus 𝑧 squared, we just wanna write it as we have because it’s more tidy when we’re dealing with fractions.
So now to actually enable us to calculate this, what we’re gonna use is another one of our operation rules for fractions. This one is the division rule. What it tells us is that if we actually have one fraction divided by another, then what we do is actually flip the second fraction, which is called the reciprocal, and then we multiply our fractions together.
So, for instance, if we had 𝑎 over 𝑏 divided by 𝑐 over 𝑑, this is gonna be equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐, which is equal to 𝑎𝑑 over 𝑏𝑐. So if we do that, we’re gonna get two-ninths multiplied by negative nine-fifths, cause we’ve actually flipped the negative five-ninths to negative nine-fifths. That’s the reciprocal.
So the next thing we do to actually make the multiplication easier is actually divide three by nine, so divide the numerator and the denominator by nine. And when we do that, we’re gonna have two over one multiplied by negative one over five. So therefore, we can say that, given that 𝑥 equal to a half, 𝑦 is equal to negative two-thirds, and 𝑧 is equal to negative a third, the numerical value of 𝑥𝑦 squared over 𝑦 plus 𝑧 squared is gonna be equal to negative two-fifths, or as a decimal it will be negative 0.4.