Simplify sqrt(40) + sqrt(250), sqrt(2) + sqrt(2) + sqrt(7), and (sqrt(6) * sqrt(30)) / sqrt(5).


Simplify {eq}\sqrt{40}+\sqrt{250},\: \sqrt{2}+\sqrt{2}+\sqrt{7},\: and\, \frac{\sqrt{6}\sqrt{30}}{\sqrt{5}}. {/eq}

Simplifying Radical Expressions:

In mathematics, simplifying radical expressions involves rewriting a radical expression in simplest form. Often, this involves manipulating the radicals within the expression to create like radicals, so that we can combine them using the operations indicated.

Answer and Explanation:

Simplifying each of the radical expressions given gives the following:

  • {eq}\sqrt{40}+\sqrt{250}=7\sqrt{10} {/eq}
  • {eq}\sqrt{2}+\sqrt{2}+\sqrt{7}=2\sqrt{2}+\sqrt{7} {/eq}
  • {eq}\frac{\sqrt{6}\sqrt{30}}{\sqrt{5}}=6 {/eq}

To simplify these expressions, we will rewrite the radicals within the expressions using the rule that {eq}\sqrt{ab}=\sqrt{a}\sqrt{b} {/eq}, then we will simplify by combining like terms and cancelling out terms, depending on the expression we are working with.

Let's start with {eq}\sqrt{40}+\sqrt{250} {/eq}. First, we are going to rewrite each of the radicals in this expression so that they become like terms. Then we will add the like terms by adding their coefficients. This goes as follows:

  • {eq}\sqrt{40}+\sqrt{250}=\sqrt{4\cdot 10}+\sqrt{25\cdot 10}=\sqrt{4}\sqrt{10}+\sqrt{25}\sqrt{10}=2\sqrt{10}+5\sqrt{10}=7\sqrt{10} {/eq}

We see that {eq}\sqrt{40}+\sqrt{250} {/eq} simplifies to {eq}7\sqrt{10} {/eq}.

Now, let's consider {eq}\sqrt{2}+\sqrt{2}+\sqrt{7} {/eq}. In this one, we can't simplify the radicals in the expression any further, but we do have two like terms within the expression, and those are {eq}\sqrt{2} {/eq} and {eq}\sqrt{2} {/eq}. To combine these two terms, we add their coefficients. When a radical has no coefficient, the implied coefficient is 1, so we have the following:

  • {eq}\sqrt{2}+\sqrt{2}+\sqrt{7}=1\sqrt{2}+1\sqrt{2}+\sqrt{7}=2\sqrt{2}+\sqrt{7} {/eq}

We get that {eq}\sqrt{2}+\sqrt{2}+\sqrt{7} {/eq} simplifies to {eq}2\sqrt{2}+\sqrt{7} {/eq}.

Lastly, let's consider the third rational expression, {eq}\frac{\sqrt{6}\sqrt{30}}{\sqrt{5}} {/eq}. To simplify this expression, we will rewrite the radicals in the numerator using our rule, and then we will perform some cancellation. This process is as follows:

  • {eq}\frac{\sqrt{6}\sqrt{30}}{\sqrt{5}}=\frac{\sqrt{6}\sqrt{6\cdot 5}}{\sqrt{5}}=\frac{\sqrt{6}\sqrt{6}\sqrt{5}}{\sqrt{5}}=\sqrt{6}\sqrt{6}=\sqrt{6\cdot 6}=\sqrt{36}=6 {/eq}

We see that {eq}\frac{\sqrt{6}\sqrt{30}}{\sqrt{5}} {/eq} simplifies to 6.

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