Question

1. where possible, replace each \\(\square\\) with a \+\ or \-\ sign to make each product positive. in other words, what sign (\+\ or \-\) would go in the box to make each expression positive when evaluated? only where possible. a) \\(-(\square 9)^{11}\\) b) \\(\square (-9)^{12}\\) c) \\(-(\square 9)^{12}\\) d) \\(\square (-9)^{11}\\)

Explanation:

Step1: Analyze part a)

We have \(-(\square 9)^{11}\). First, recall that for an odd exponent \(n\), \((-a)^n=-a^n\) and \((a)^n = a^n\). Let's consider the base \((\square 9)\). The exponent \(11\) is odd. We want the whole expression \(-(\square 9)^{11}\) to be positive. Let's denote \(x=\square 9\), so the expression is \(-x^{11}\). We want \(-x^{11}>0\), which implies \(x^{11}<0\). For \(x^{11}<0\), \(x\) must be negative (since odd exponent preserves the sign). So \(x = - 9\), which means \(\square\) should be \(-\) because \(-9\) is the base. Let's check: \(-(-9)^{11}=-(-9^{11}) = 9^{11}\), which is positive.

Step2: Analyze part b)

We have \(\square(-9)^{12}\). The exponent \(12\) is even, so \((-9)^{12}=9^{12}\) (positive). We want \(\square\times9^{12}\) to be positive. Since \(9^{12}\) is positive, the sign of \(\square\) must be positive (because positive times positive is positive). So \(\square\) should be \(+\). Check: \(+(-9)^{12}=9^{12}\), positive.

Step3: Analyze part c)

We have \(-(\square 9)^{12}\). The exponent \(12\) is even, so \((\square 9)^{12}\) is always non - negative (in fact, positive if \(\square 9
eq0\)). Let \(x = \square 9\), then the expression is \(-x^{12}\). Since \(x^{12}\geq0\), then \(-x^{12}\leq0\). The only way \(-x^{12}\) is positive is if \(x^{12}<0\), but any real number raised to an even power is non - negative. So it's not possible to make this expression positive.

Step4: Analyze part d)

We have \(\square(-9)^{11}\). The exponent \(11\) is odd, so \((-9)^{11}=-9^{11}\) (negative). We want \(\square\times(-9^{11})\) to be positive. Since \(-9^{11}\) is negative, the sign of \(\square\) must be negative (because negative times negative is positive). Let's check: \(-(-9)^{11}=-(-9^{11}) = 9^{11}\), which is positive.

Answer:

a) \(\square=- \)

b) \(\square=+\)

c) Not possible

d) \(\square=-\)