Video: Evaluating Algebraic Expressions Using the Properties of Square Roots

Given that π‘₯ = (2√5 + √2)/2√7 and 𝑦 =(√5 βˆ’ 3√2)/√42, find the value of π‘₯Β² + 𝑦².

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Video Transcript

Given that π‘₯ is equal to two root five plus root two divided by two root seven and 𝑦 is equal to root five minus three root two divided by root 42, find the value of π‘₯ squared plus 𝑦 squared.

In order to solve this problem, we’ll find an expression for π‘₯ squared and an expression for 𝑦 squared, and then add our two answers. As π‘₯ is equal to two root five plus root two divided by two root seven, then π‘₯ squared must be equal to two root five plus two divided by two root seven multiplied by two root five plus root two divided by two root seven as squaring an expression involves multiplying it by itself. When we are multiplying two fractions, we multiply the two numerators together and the two denominators together.

We will expand the two brackets on the top using the FOIL method. Multiplying the first terms two root five multiplied by two root five gives us 20 as two multiplied by two is equal to four and root five multiplied by root five is equal to five; four multiplied by five gives us 20. Multiplying the outside terms, two root five and root two, gives us two root 10 as root five multiplied by root two is equal to root 10. Multiplying the inside terms, root two and two root five, also gives us two root 10. And finally, multiplying the last terms, root two multiplied by root two, gives us two.

Multiplying the denominator, two root seven multiplied by two root seven, gives us 28 as two multiplied by two is equal to four and root seven multiplied by root seven is equal to seven; four multiplied by seven is equal to 28. Simplifying the numerator by collecting like terms gives us 22 plus four root 10 divided by 28. We can simplify this fraction by dividing the numerator and denominator by two. Whatever you do to the top, you must do the bottom. Dividing the numerator by two gives us 11 plus two root 10 and dividing the denominator by two gives us 14. We can therefore say that π‘₯ squared is equal to 11 plus two root 10 divided by 14.

We now need to repeat this process for 𝑦. 𝑦 squared is equal to root five minus three root two divided by root 42 multiplied by root five minus three root two divided by root 42. Once again, we’ll multiply the two numerators and then, separately, the two denominators. We’ll once again expand the numerators using the FOIL method. Multiplying the first terms, root five by root five, gives us five. Multiplying the outside terms gives us negative three root 10 as root five multiplied by root two is root 10. Multiplying the inside terms also gives us negative three root 10. Finally, multiplying the last terms gives us 18. Three multiplied by three is equal to nine, and root two multiplied by root two is equal to two; nine times two equals 18.

It’s also important to note here that multiplying two negative numbers give us a positive answer. Multiplying the denominators, root 42 by root 42, gives us 42. We can then simplify the numerator by collecting like terms: five plus 18 is equal to 23, and negative three root 10 minus three root 10 is equal to negative six root 10. Therefore, 𝑦 squared equals 23 minus six root 10 divided by 42. This cannot be simplified. So we now have expressions for π‘₯ squared and 𝑦 squared. The question asked us to add the expression for π‘₯ squared to the expression for 𝑦 squared. We therefore have 11 plus two root 10 divided by 14 plus 23 minus six root 10 divided by 42.

In order to add or subtract fractions, we need to make sure that the denominators are the same. In this case, we can multiply the numerator and denominator of the first fraction by three, giving us a common denominator of 42 as 14 multiplied by three is equal to 42. Multiplying the numerator of the first fraction by three gives us 33 plus six root 10, and multiplying the denominator by three gives us 42. Once we have a common denominator, we just need to add the two numerators: 33 plus 23 is equal to 56 and six root 10 minus six root 10 is equal to zero. The denominator remains as 42. Therefore, π‘₯ squared plus 𝑦 squared is equal to 56 over 42 or 56 divided by 42.

We can simplify this fraction by dividing by a common factor. The highest common factor of 42 and 56 is 14. 56 divided by 14 is equal to four as four multiplied by 14 equals 56. 42 divided by 14 equals three as three multiplied by 14 is equal to 42. This means that our value for π‘₯ squared plus 𝑦 squared can be simplified to four-thirds or four over three. If π‘₯ is equal to two root five plus two divided by two root seven and 𝑦 is equal to root five minus three root two divided by root 42, then π‘₯ squared plus 𝑦 squared equals four-thirds.

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