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Question Video: Performing Numeric Fraction Multiplications to Determine Whether an Expression Yields a Result Which is Rational Mathematics • 7th Grade

If π‘Ž = 3, 𝑏 = βˆ’2, 𝑐 = 4, and 𝑑 = βˆ’1, which of the following expressions results in a rational number? [A] ((π‘Ž + 𝑏 βˆ’ 𝑐)/𝑏) Γ— ((π‘Ž + 𝑏 + 𝑐)/(2𝑑 βˆ’ 𝑏)) [B] π‘Ž/(𝑑 + (𝑐 βˆ’ π‘Ž)) Γ— 𝑏/𝑐 [C] βˆšπ‘Ž/π‘Ž Γ— βˆšπ‘/𝑐 [D] ((π‘Ž + 𝑏 + 𝑐)/(𝑏 + π‘Ž)) Γ— ((π‘Ž + 𝑏 + c)/(2𝑑 βˆ’ 𝑏)) [E] (π‘Ž + 𝑏)/𝑏 Γ— (𝑐 + 𝑑)/𝑑

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Video Transcript

If π‘Ž is equal to three, 𝑏 is equal to negative two, 𝑐 is equal to four, and 𝑑 is equal to negative one, which of the following expressions results in a rational number? Option (A) π‘Ž plus 𝑏 minus 𝑐 all over 𝑏 multiplied by π‘Ž plus 𝑏 plus 𝑐 over two 𝑑 minus 𝑏. Option (B) π‘Ž over 𝑑 plus 𝑐 minus π‘Ž multiplied by 𝑏 over 𝑐. Option (C) root π‘Ž over π‘Ž times root 𝑐 over 𝑐. Option (D) π‘Ž plus 𝑏 plus 𝑐 all over 𝑏 plus π‘Ž multiplied by π‘Ž plus 𝑏 plus c over two 𝑑 minus 𝑏. Or is it option (E) π‘Ž plus 𝑏 all over 𝑏 multiplied by 𝑐 plus 𝑑 over 𝑑?

In this question, we are given five expressions involving given constants π‘Ž, 𝑏, 𝑐, and 𝑑. And we need to determine which expression results in a rational number. We can start by recalling that a rational number is the quotient of any two integers where the denominator is nonzero. Therefore, we can answer this question by evaluating each of the expressions by substituting the given values of these constants into the expression and seeing which expression is the quotient of two integers with a nonzero denominator.

We can simplify this process slightly by noting that each of the expressions involves a product. And we know that the product of any two rational numbers is rational by the closure property of the product of rational numbers. So, we do not need to calculate the full expression if we have the product of two rational numbers. We can conclude that their product is rational.

Let’s start by evaluating the expression in option (A). Substituting the values of π‘Ž, 𝑏, 𝑐, and 𝑑 into the expression gives us three plus negative two minus four all over negative two multiplied by three plus negative two plus four over two times negative one minus negative two. We could evaluate each expression in the numerators and denominators. However, it is useful to note that these expressions are the product and sum of integers.

So by the closure properties, these are all integers. Therefore, we do not need to evaluate the numerators since these are already integers, we only need to check that our final answer has a denominator that is nonzero. However, we can calculate that two times negative one minus negative two is zero. We cannot divide by zero. So, this expression is not defined for these values of π‘Ž, 𝑏, 𝑐, and 𝑑. And so, it is not a rational number.

We can apply a similar process to option (B). We note that both factors in this expression are the quotients of integers. So, we just need to check that we are not dividing by zero. Substituting the values of π‘Ž, 𝑏, 𝑐, and 𝑑 into the expression gives us the following. We can evaluate the expression in the first denominator to see that negative one plus four minus three is equal to zero. So, this expression is undefined. Hence, it is not a rational number.

The expression in option (C) gives us a different problem. If we substitute the values of π‘Ž and 𝑐 into the expression, we obtain root three over three times root four over four. We can start by noting that four is a perfect square. So, the square root of four is equal to two. We can then simplify two over four to give us one half. However, we cannot write this expression as the quotient of two integers. The reason for this is that there is no integer whose square is three. In other words, three is not a perfect square.

In option (D), we can once again note that both of the numerators and denominators are integers. So, we only need to check that we are not dividing by zero. However, we have already seen that two 𝑑 minus 𝑏 is equal to zero. So, this expression is undefined and not a rational number.

This leaves us with option (E). We see that we are multiplying two fractions. And once again, we can note that these are both the quotients of integers. We can then note that both denominators are nonzero, since 𝑏 is negative two and 𝑑 is negative one. Therefore, both of the factors in this expression are rational numbers. So, this is the product of two rational numbers.

Hence, by the closure property of the multiplication of rational numbers, we can conclude that the expression in option (E) results in a rational number.

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