Question Video: Evaluating an Expression Using the Order of Operations | Nagwa Question Video: Evaluating an Expression Using the Order of Operations | Nagwa

Question Video: Evaluating an Expression Using the Order of Operations Mathematics • First Year of Preparatory School

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Given that π‘₯ = 1/2, 𝑦 = 2/3, and 𝑧 = βˆ’2/5, find the value of (π‘₯ βˆ’ 𝑦)/𝑧².

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Video Transcript

Given that π‘₯ is equal to one-half, 𝑦 is equal to two-thirds, and 𝑧 is equal to negative two-fifths, find the value of π‘₯ minus 𝑦 all over 𝑧 squared.

In this question, we are given rational numbers π‘₯, 𝑦, and 𝑧 and asked to evaluate an algebraic expression involving π‘₯, 𝑦, and 𝑧. The first thing that we can note about the expression that we are given is that it contains fraction notation. This means that we want to evaluate the numerator and denominator separately before the division.

We can think of this as having parentheses around the numerator and denominator. However, for simplicity, these are usually left out. To evaluate the expression, we need to substitute the values of π‘₯, 𝑦, and 𝑧 into the expression. This gives us that π‘₯ minus 𝑦 all over 𝑧 squared is equal to one-half minus two-thirds all over negative two-fifths squared.

We now want to evaluate the expressions in the numerator and denominator. Let’s start with the numerator. This is the difference between two rational numbers. So we want their denominators to be equal. The lowest common multiple of the denominators is six. So we rewrite the first rational number as three over six and the second rational number as four over six to obtain the following difference. We can then evaluate the difference between these fractions by finding the difference in their numerators. Since three minus four is equal to negative one, we get negative one over six. This is the numerator in our expression.

We now need to evaluate the denominator. To do this, we can recall that we can square a fraction by squaring its numerator and denominator separately. Therefore, negative two over five all squared is equal to negative two squared over five squared, which we can calculate is equal to four over 25. Hence, the denominator of our expression is four over 25. For clarity, we will add the parentheses in the numerator and denominator. We see that π‘₯ minus 𝑦 all over 𝑧 squared is equal to negative one-sixth over four over 25.

Now, our expression is the quotient of two rational numbers. And we can recall that dividing by a nonzero fraction is the same as multiplying by its reciprocal. So, π‘Ž over 𝑏 divided by 𝑐 over 𝑑 is equal to π‘Žπ‘‘ over 𝑏𝑐 provided 𝑏, 𝑐, and 𝑑 are nonzero. In our case, our value of π‘Ž is negative one, 𝑏 is six, 𝑐 is four, and 𝑑 is 25. So we obtain negative one times 25 over six times four.

Finally, we can evaluate these products to get negative 25 over 24. We can note that we cannot simplify this fraction any further. Hence, our answer is that π‘₯ minus 𝑦 all over 𝑧 squared is equal to negative 25 over 24.

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