Video Transcript
Simplify four root 10 minus three root seven over four root 10 by rationalizing the denominator.
We’ve been asked to simplify this fraction by rationalizing its denominator, which means we need to rewrite the fraction in a form where its denominator is a rational number. We do this by multiplying both the numerator and denominator by a value that is carefully chosen to ensure its product with the denominator gives a rational value. Because we’re multiplying both the numerator and denominator of the fraction by the same value, the result is equivalent to the original fraction.
When the denominator is the product of a rational number and a square root as we have here, the value we multiply by is just the square root itself. This is because for any nonnegative value 𝑎, the square root of 𝑎 multiplied by the square root of 𝑎 is equal to 𝑎. Multiplying by root 10 over root 10, then, gives root 10 multiplied by four root 10 minus three root seven over four root 10 multiplied by root 10. In the denominator, root 10 multiplied by root 10 gives 10 and then multiplying by four gives 40. So the denominator is now a rational number.
Distributing the parentheses in the numerator gives four times 10 for the first term, as root 10 multiplied by root 10 is 10. And then we subtract three root 70 for the second term. Here we are also applying the result that for nonnegative values of 𝑎 and 𝑏, root 𝑎 multiplied by root 𝑏 is equal to the square root of 𝑎𝑏. So root seven multiplied by root 10 is root 70.
Finally, we simplify by evaluating the product of four and 10. This fraction can’t be simplified any further, as firstly 70 has no square factors other than one. So the radical cannot be simplified. And secondly, the three values of 40, three, and 40 don’t all share any common factors other than one.
Our answer is that the simplified form of the given quotient with a rational denominator is 40 minus three root 70 over 40.
