Calculate the square root of 12 over five squared minus 16 times 12 over five plus 64.
To do this, we’ll need to remember the order of operations. We sometimes use PEMDAS to help us remember. Parentheses, exponents, multiplication and division from left to right, and then addition and subtraction from left to right. Multiplication and division are in the same step. And addition and subtraction are in the same step. And even though we use a P and say parenthesis here, this step could be brackets or any other form of grouping.
So, let’s consider our expression. We’re taking the square root of everything under the radical, the square root of 12 over five squared minus 16 times 12 over five plus 64. This means we’ll need to calculate everything under the radical first. Inside the radical, we’ll still follow the order of operations. We don’t have any other grouping under the radical, and so we can proceed to the exponents.
We have 12 over five squared. We can rewrite that to say 12 squared over five squared. 12 squared equals 144. Five squared equals 25. And that completes the exponents step under the radical. Next, we’ll do any multiplication or division from left to right. We need to multiply 16 by twelve-fifths. To do that, we multiply 16 by 12 in the numerator and the denominator stays the same, five. 16 times 12 is 192. Bring down our sign. Bring down the subtraction. This finishes our multiplication and division step.
And so, we bring down the plus 64. We now need to add or subtract underneath the radical from left to right. Notice that we’re dealing with fractions that do not have the same denominator. Before we can add and subtract these values, we need to find a common denominator. If we multiply five times five, we get 25. And if we multiply by five in the denominator, we must multiply by five in the numerator.
Now, we have two fractions that have a denominator of 25. And then, we could multiply 64 by 25 over 25. We would have 144 over 25 minus 960 over 25 plus 1600 over 25. Now that these fractions have a common denominator, we can add and subtract. Which would give us 144 minus 960 plus 1600 over 25. 144 minus 960 plus 1600 equals 784. And the denominator stays the same, 25. We’re now finished with our addition and subtraction underneath the radical.
To simplify this further, we can break up this radical into two pieces, the square root of 784 over the square root of 25. 784 is a square number. 28 times 28 equals 784. And that means the square root of 784 is 28. We remember that there is both a positive and a negative solution. Negative 28 times negative 28 also equals 784. 25 is also a square number. The square root of 25 is plus or minus five.
This means we could have positive 28 over positive five, negative 28 over negative five, which would be equal to 28 over five. Positive 28 over negative five, which would be negative twenty-eight fifths. Or negative 28 over positive five, which would again be negative twenty-eight fifths. And so, we would write this as plus or minus 28 over five, plus or minus twenty-eight fifths.