Video Transcript
Calculate the numerical value of five over four to the power of 𝑦 multiplied by five over four to the power of 𝑥 when 𝑥 equals negative seven and 𝑦 equals four.
So to solve this problem, what we’re going to do is substitute in our 𝑥- and 𝑦-values into the expression that we have. So when we do that, what we’re gonna get is five over four or five-quarters all to the power of four multiplied by five over four all to the power of negative seven. Well, the key here in this question is not to straightaway try and work out what the numerical value of each of the terms is. So we don’t straightaway go write five over four to the power of four — well, we put five to the power of four, four to the power of four — and work it out that way because that’d be very long winded and complicated.
So what we do need to do is notice that with both of our terms, we have the same base number. And if we have the same base number, then what we can use is one of our index or exponent rules. And what that rule states is that if we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏, this is equal to 𝑥 to the power of 𝑎 plus 𝑏. So what we do is we add the powers or add the exponents. So therefore, what we’re gonna have in our example is five over four all to the power of, and then we’ve got four plus negative seven because we’ve added together the exponents. Or if you add a negative, it’s the same as subtracting a positive.
So then what we’re gonna have is five over four to the power of negative three. But what do we do now? So what we have is another exponent rule. And that tells us if we have 𝑥 to the power of negative 𝑎, this is equal to one over 𝑥 to the power of 𝑎. So what we can see is this is in fact the reciprocal. And then we have 𝑥 to the power of 𝑎. Well, what we know about reciprocal is that if we have the fraction five over four or five-quarters, then the reciprocal of this is four over five.
So now, using our exponent rule, we’re gonna have four over five to the power of three. Okay, finally. Now, what’s the next step? Well, we want to find the numerical value, so we want to work out what four over five or four-fifths to the power of three is. And to do that, what we do is we utilize one more exponent rule. And that is that if we have a fraction, so 𝑥 over 𝑦, and then this is all to the power of 𝑎, then this is equal to 𝑥 to the power of 𝑎 over 𝑦 to the power of 𝑎.
So what we’re gonna get is four cubed over five cubed. So finally, what we’re gonna get is a numerical value of five over four to the power of 𝑦 multiplied by five over four to the power of 𝑥 when 𝑥 equals negative seven and 𝑦 equals four, which is 64 over 125. And we get that cause four cubed is 64 and five cubed is 125.
