Question Video: Evaluating Numerical Expressions Involving Square Roots Mathematics • 8th Grade

Calculate √(1(7/9)) βˆ’ 19/32.

02:44

Video Transcript

Calculate the square root of one and seven-ninths minus 19 over 32.

In this question, we are asked to evaluate an expression involving the square root of a mixed number and then the difference of this result with a rational number.

To do this, we first recall that the square root of a number π‘Ž is the nonnegative number whose square is π‘Ž. It is easier to find the square root of a fraction than a mixed number. So we will convert one and seven-ninths into a fraction by rewriting one as nine over nine. We obtain the square root of 16 over nine minus 19 over 32.

Now that we have the square root of a fraction, we can recall that quotient rule for square roots tells us that if π‘Ž and 𝑏 are integers with π‘Ž nonnegative and 𝑏 positive, then the square root of π‘Ž over 𝑏 is equal to the square root of π‘Ž over the square root of 𝑏. Applying this result with π‘Ž equal to 16 and 𝑏 equal to nine gives us the square root of 16 over the square root of nine minus 19 over 32.

We can now evaluate both of the square roots by recalling that if 𝑐 is a nonnegative number, then the square root of 𝑐 squared is just equal to 𝑐. Therefore, since 16 is equal to four squared and nine is equal to three squared, we can evaluate the square roots to get four-thirds minus 19 over 32.

We now need to find the difference between the two fractions. And to do this, we need them to have the same denominator. We can find that the lowest common multiple of the denominators is 96. So we will multiply the numerator and denominator of the first fraction by 32 and the numerator and denominator of the second fraction by three. This gives us 128 over 96 minus 57 over 96, which we can calculate is equal to 71 over 96. We cannot simplify this fraction any further, so our final answer is 71 over 96.

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