Video: Simplifying Numerical Expressions Using the Properties of Square Roots

Express √15(11√5 + 2√3) in its simplest form.

02:43

Video Transcript

Express root 15 multiplied by 11 root five plus two root three in its simplest form.

In order to simplify this expression, we need to use some of the laws of surds. Root 𝑎 multiplied by root 𝑎 is equal to 𝑎. Root 𝑎 multiplied by root 𝑏 is equal to root of 𝑎𝑏. And root 𝑎 multiplied by 𝑏 is equal to 𝑏 root 𝑎.

In order to expand or multiply out the parenthesis, we need to multiply root 15 by 11 root five and then root 15 by two root three. Let’s first consider root 15 multiplied by 11 root five. Root 15 can be split into root three multiplied by root five, as three multiplied by five is 15. 11 root five can be rewritten as 11 multiplied by root five.

Using the law of surds root 𝑎 multiplied by root 𝑎 is equal to 𝑎, we can see that root five multiplied by root five is equal to five. This gives us root three multiplied by five multiplied by 11. As five multiplied by 11 is 55, the first expression becomes 55 root three. Root 15 multiplied by 11 root five is 55 root three.

Now let’s consider the second part of the expansion, root 15 multiplied by two root three. If we split the two terms in the same way as the first part, we’re left with root three multiplied by root five multiplied by two multiplied by root three. Root three multiplied by root three is equal to three. So we now have three multiplied by root five multiplied by two. This is equal to six root five.

Therefore, root 15 multiplied by two root three is equal to six root five. Root 15 multiplied by 11 root five plus two root three in its simplest form is written 55 root three plus six root five.

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