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In this video, we'll be discussing simplifying square roots and rationalizing denominators.
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A square root is simplified when the radicand has no perfect square factors other than one.
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For example, the square root of twenty is a radical expression that is not simplified
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because I can write the radicand twenty as the product of two factors one of which is a perfect
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square other than one. Twenty is the same as four times five. So the square root of twenty
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can be written as the square root of four times five, and four is a perfect square.
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The Product Rule for Square Roots says that if the square root of a and the square root of b
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are real numbers, then the square root of a times b is equal to the square root of a times
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the square root of b. We can use this product rule to continue simplifying the square root of twenty.
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Now that we have rewritten the radicand as four times five, we can use the product rule
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to separate this radical expression into two radical factors, the square root of four times
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the square root of five. And we know that the square root of four is two. So this gives us two
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times the square root of five. Now the radicand five only has factors of one and five, so it
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is simplified. Again, as long as the radicand has no perfect square factors other than one,
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it is simplified. Two root five is equivalent to root twenty, and it is in the simplified form.
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Now let's look at some examples. Here we want to simplify these radical expressions. We'll begin with the
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square root of fifty-four. We need to think of two numbers that can multiply together to give us fifty-four, one of which is a
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perfect square, and ideally, we'd like to choose the largest perfect square that divides into
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this radicand. So let's think of the ways that we can multiply two numbers to get fifty-four.
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Well, we can multiply nine and six to get fifty-four, and nine is a perfect square.
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If we tried other factors, for example two and twenty-seven, those are factors of fifty-four, but
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two is not a perfect square nor is twenty-seven. Now let's use the product rule to simplify this.
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The square root of nine times six can be written as the square root of nine
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times the square root of six, and the square root of nine is three. This gives us three
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times the square root of six. Now ask yourself are there any perfect square factors other
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than one that divide evenly into the radicand six. There are not. So we are simplified here. The next expression is the
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square root of five hundred. Again, we're looking for two numbers that multiply together to give me five hundred.
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One of them needs to be a perfect square, and we'd like to choose the largest perfect square
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that divides evenly into five hundred. That's one hundred. Five hundred can be written as
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one hundred times five. So the square root of five hundred can be written as the square root
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of one hundred times five which is the square root of one hundred times the square root of five.
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The square root of one hundred is ten, so this gives us ten times
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the square root of five. We know that we are completely simplified
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when the radicand has no perfect square factors other than one.
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And let's look at our third example. We have three times the square root of twelve. The radicand is
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twelve. Twelve can be broken down into two factors, one of which is a perfect square.
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Three times four is twelve. So we can write this as three times the square root of four times
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three, and this can be rewritten using the product rule as three times the square root of four
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times the square root of three. The square root of four is two. So this gives us three times two
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times the square root of three. And our final step here will be multiplying the two factors that are
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outside of the radical. Three times two is six, and so this gives us six times the square root of
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three as our simplified radical expression that is equivalent to three square root of twelve.
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Next, we're going to examine the square root of a quotient. The Quotient Rule
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for Square Roots says that if a and b are real numbers and b is not equal to zero,
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then the square root of the quotient a divided by b is equal to the square root of a
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divided by the square root of b. Notice that b can't be zero because then you would have zero
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in the denominator of a fraction which is undefined. Let's work a couple of examples.
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We'll begin by simplifying the square root of five over forty-nine, five forty-ninths. Let's rewrite
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this using the quotient rule as the square root of five divided by the square root of forty-nine.
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And now let's see if either of these can be simplified or evaluated.
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The square root of five cannot be simplified. There are no factors
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other than one that are perfect squares of five. The square root of forty-nine is seven.
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So this can be written as the square root of five over seven. In our next example, we have the square root
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of one over one twenty-one. Using the quotient rule, we can rewrite this
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as the square root of one divided by the square root of one twenty-one. Now you may
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notice that both the numerator and denominator are perfect squares, and each can be evaluated.
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The square root of one is one, and the square root of one twenty-one is
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eleven. So the square root of one over one twenty-one is one-eleventh.
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Sometimes it is easier to work with a radical expression if the denominator does not contain
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a radical. Rewriting a radical expression to eliminate a radical in the denominator
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is called rationalizing the denominator. For example, the square root of five divided by
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the square root of two. We can see that in this fraction we have a radical expression
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both in the numerator and the denominator. If we would like to rationalize the denominator,
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meaning that we want a rational value as opposed to an irrational number in the denominator,
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then we need a way to eliminate this radical expression in the denominator.
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What could I multiply this radicand by such that the value in the denominator would be a rational
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value? We know that four is a perfect square. So the square root of four would give us a rational
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value. In order to accomplish that, we would multiply the denominator
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by the square root of two, and if I multiply the denominator of a fraction by the square
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root of two, I must also multiply the numerator by the square root of two
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so that I'm not changing the value of this expression. You can see here that all I'm
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doing is multiplying by one. Now using a version of the product rule for radicals
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in the opposite direction of what we used it in the previous examples, I can combine these
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two radicands under a single radical sign. So the square root of two times the square root of
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two gives me the square root of two times two which we know to be the square root of four.
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And the numerator, the square root of five times the square root of two,
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gives us the square root of five times two which gives us the square root of ten.
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And now we can evaluate this denominator, the square root of four,
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which is two. This gives us the square root of ten over two. The square root of ten in the numerator
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is a simplified radical expression. It has no factors that are perfect squares other than one.
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And the denominator is a rational number. So we have rationalized the denominator
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of our original problem. These expressions, the square root of five over the square root of two
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and the square root of ten over two, are equivalent expressions. Let's work a
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few examples. We want to rationalize the denominator of these expressions and simplify.
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We'll begin with the square root of thirteen divided by the square root of
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fifteen. The denominator of this fraction is an irrational number, the square root of fifteen.
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We want to be able to have a rational number, so we will multiply by the square root of fifteen in
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the denominator which means we must also multiply by the square root of fifteen in the numerator.
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Using the product rule now allows us to combine these radicands.
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The denominator is the square root of fifteen times fifteen which can be written as the square
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root of fifteen squared, and the numerator is the square root of thirteen times fifteen.
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The square root of thirteen times the square root of fifteen is the square root
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of one hundred ninety-five, and the square root of fifteen squared, well, that's just fifteen. And now
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we have a rational denominator. The square root of one ninety-five over fifteen
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is equivalent to the original problem square root of thirteen divided by square root of fifteen.
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And in our last example, we have one divided by the square root of seventy-two.
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The square root of seventy-two is an irrational number in the denominator of our fraction.
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We have two options in simplifying this expression. We could multiply
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both the numerator and denominator by root seventy-two to rationalize this denominator,
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or we can simplify the square root of seventy-two first which will make our
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multiplication a little bit more efficient because the numbers will be smaller.
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Let's do that. Let's begin by simplifying the square root of seventy-two.
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What two numbers multiply together to give me seventy-two, one of which is a perfect square? And
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ideally we'd like to choose the largest perfect square that divides evenly into seventy-two.
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Well, those numbers would be thirty-six and two. So we'll rewrite the square root of seventy-two
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as the square root of thirty-six times two. Now using our product rule, we know that this
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can be written as the square root of thirty-six times the square root of two
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And the square root of thirty-six is six. That gives us six times the square root of two.
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So our denominator is simplified, but it is still an irrational number.
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So to rationalize this denominator, we want this radical expression or the radicand rather to be a
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perfect square. So we will multiply the numerator and denominator by the square root of two.
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Now using the product rule, we know that the square root of two
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times the square root of two is the square root of four, and the square root of four is two. That
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gives us six times two in our denominator. And one times the square root of two in the numerator
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is just the square root of two. So our final simplified answer will be the square root of two
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divided by twelve. The square root of two divided by twelve has a rational value in the denominator,
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and it is equivalent to one over the square root of seventy-two.