Question 635765
First a quick review.<ul><li>Every positive number has <u>two</u> square roots, a positive square root and a negative square root. For example, 25 has 5 and -5 as square roots.</li><li>The square root symbol is used <u>only</u> to express the positive square root. To express the negative square root you put a minus in front of the square root symbol. To express both square roots you put the "plus or minus" symbol in front. For example:
{{{sqrt(25) = 5}}} 
{{{-sqrt(25) = -5}}}
<u>+</u>{{{sqrt(25)}}} refers to both 5 and -5.</li><li>The expression inside the square root symbol, called the radicand, is the result of squaring the square root. For example, in {{{sqrt(25)}}} the radicand is 25 and it is the result of squaring {{{sqrt(25)}}},</li></ul>Now let's look at your square root:
{{{sqrt((a+7)^2)}}}
First, since there is no "-" or "<u>+</u>" in front this is a reference to the positive square root. So whatever we end up with for an answer, it <u>must</u> be positive.<br>
Next, this square root represents the number/expression you square to get {{{(a+7)^2}}} as a result. So what do we square to get {{{(a+7)^2}}}? The "obvious" answer is: a+7.<br>
But here is where we can easily go wrong. We can't forget the part about "must be positive". Is a+7 guaranteed to be positive? Since the problem tells us that the variable can be any real number, we must say: No, a+7 is not necessarily positive. So what do we do? Answer: Use absolute value to guarantee a positive square root:
{{{sqrt((a+7)^2) = abs(a+7)}}}<br>
Note: Use the absolute value on the whole expression, not just the variable. {{{abs(a)+7}}} is not correct.