Integer Exponents - Accuplacer Qas

What is the value of $2^{-1} + 4^{-1}$ ?
Answer: $\frac{3}{4}$ The expression must be evaluated term by term. The negative exponent indicates a reciprocal. So, $2^{-1} = \frac{1}{2}$ and $4^{-1} = \frac{1}{4}$ . The problem becomes $\frac{1}{2} + \frac{1}{4}$ . To add these fractions, find a common denominator, which is 4. $\frac{1}{2} = \frac{2}{4}$ . So, the sum is $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$ . * A is incorrect. This may result from misinterpreting negative exponents as making the numbers negative. * B is incorrect. This is a common error from adding the denominators ( $2+4=6$ ) or from incorrectly adding the bases first, $(2+4)^{-1} = 6^{-1} = 1/6$ . * C is incorrect. This can result from the common fraction addition error of adding numerators and denominators: $\frac{1+1}{2+4} = \frac{2}{6} = \frac{1}{3}$ .
If $x \neq 0$ , what is the value of the expression $4x^0 - (2x)^0$ ?
Answer: 3 Any non-zero quantity raised to the power of 0 is equal to 1. The expression has two terms. In the first term, $4x^0$ , only the $x$ is raised to the power of 0. So, $4x^0 = 4(1) = 4$ . In the second term, $(2x)^0$ , the entire quantity $2x$ is raised to the power of 0, so $(2x)^0 = 1$ . The expression simplifies to $4 - 1 = 3$ . * A is incorrect. This result might come from incorrectly evaluating $4x^0$ as $(4x)^0=1$ , leading to $1-1=0$ . * B is incorrect. This result might come from misinterpreting the parentheses, as in $4x^0 - 2x^0 = 2x^0 = 2(1) = 2$ . * D is incorrect. This would happen if the second term, $(2x)^0$ , was incorrectly evaluated as 0.
The number of cells in a culture triples every hour. If there are $N$ cells at 12:00 PM, which expression represents the number of cells that were in the culture at 9:00 AM on the same day?
Answer: $3^{-3}N$ The time 9:00 AM is 3 hours before 12:00 PM. The population is modeled by $N \cdot 3^t$ , where $t$ is the number of hours after 12:00 PM. To find the population 3 hours before 12:00 PM, we must use $t = -3$ . So the expression is $N \cdot 3^{-3}$ , which can also be written as $\frac{N}{3^3}$ or $\frac{N}{27}$ . * B is incorrect. This expression represents the population 3 hours after 12:00 PM, at 3:00 PM. * C is incorrect. This represents the population one hour prior, at 11:00 AM, not three hours prior. * D is incorrect. This represents a linear decrease, but the cell growth is exponential.
If $a = -3$ , what is the value of the expression $-a^2 - a^3$ ?
Answer: 18 Substitute $a = -3$ into the expression: $-(-3)^2 - (-3)^3$ . Following the order of operations, evaluate the exponents first. For the first term, $(-3)^2 = (-3)(-3) = 9$ . The term becomes $-(9)$ , or $-9$ . For the second term, $(-3)^3 = (-3)(-3)(-3) = -27$ . The expression is now $-9 - (-27)$ , which simplifies to $-9 + 27 = 18$ . * A is incorrect. This results from incorrectly evaluating $(-3)^3$ as $27$ , leading to $-9 - 27 = -36$ . * B is incorrect. This results from making sign errors on both terms, possibly calculating $-(-3)^2$ as $9$ and $(-3)^3$ as $27$ , leading to $9-27=-18$ . * D is incorrect. This results from a common error in the first term, calculating $-a^2$ as $(-a)^2$ , which would be $(-(-3))^2 = 3^2 = 9$ . This leads to $9 - (-27) = 36$ .
If $a^{-3} = \frac{1}{8}$ and $b^2 = 9$ , what is the value of $(ab)^{-1}$ ?
Answer: $\pm \frac{1}{6}$ From $a^{-3} = \frac{1}{8}$ , we get $\frac{1}{a^3} = \frac{1}{8}$ , so $a^3 = 8$ , which means $a = 2$ . From $b^2 = 9$ , we get $b = \pm 3$ . Therefore, $ab = 2(\pm 3) = \pm 6$ , and $(ab)^{-1} = \frac{1}{ab} = \frac{1}{\pm 6} = \pm \frac{1}{6}$ . Choice B would result from considering only the positive value of $b$ . Choice C results from incorrectly computing $a = \frac{1}{2}$ from the given condition. Choice D results from both using only positive $b$ and the incorrect value of $a$ .
Which of the following is equivalent to $\frac{2a^{-3}b^4}{6a^2b^{-1}}$ for $a \neq 0, b \neq 0$ ?
Answer: $\frac{b^5}{3a^5}$ Simplify the expression in parts: coefficients, $a$ terms, and $b$ terms. The coefficients $\frac{2}{6}$ simplify to $\frac{1}{3}$ . For the variables, use the quotient rule $\frac{x^m}{x^n} = x^{m-n}$ . For $a$ , we have $\frac{a^{-3}}{a^2} = a^{-3-2} = a^{-5}$ . For $b$ , we have $\frac{b^4}{b^{-1}} = b^{4-(-1)} = b^{4+1} = b^5$ . Combining these parts gives $\frac{1}{3}a^{-5}b^5$ . Since $a^{-5} = \frac{1}{a^5}$ , the expression is $\frac{b^5}{3a^5}$ . * A is incorrect. This results from adding exponents instead of subtracting ( $a^{-3+2}=a^{-1}$ ) and subtracting exponents incorrectly ( $b^{4-1}=b^3$ ). * B is incorrect. This results from subtracting exponents for $b$ incorrectly ( $b^{4-1}=b^3$ ). * D is incorrect. This results from an error in simplifying the coefficients ( $6/2=3$ instead of $2/6=1/3$ ).
Which of the following expressions is equivalent to $(2x^3)^2(x^{-1})$ for $x \neq 0$ ?
Answer: $4x^5$ First, evaluate $(2x^3)^2$ . The exponent 2 applies to both the coefficient 2 and the variable $x^3$ . This gives $2^2(x^3)^2 = 4x^{3 \cdot 2} = 4x^6$ . Now, multiply this result by $x^{-1}$ : $4x^6 \cdot x^{-1}$ . Using the product of powers rule ( $a^m \cdot a^n = a^{m+n}$ ), we get $4x^{6+(-1)} = 4x^5$ . * A is incorrect. This results from forgetting to square the coefficient 2. * C is incorrect. This is the intermediate result before multiplying by $x^{-1}$ . * D is incorrect. This results from incorrectly adding the exponents, perhaps by making a sign error: $4x^{6+1} = 4x^7$ .
If $3^x = 81$ and $2^y = 64$ , what is the value of $x-y$ ?
Answer: -2 First, find the value of $x$ . We need to determine what power of 3 equals 81. $3^1=3$ , $3^2=9$ , $3^3=27$ , $3^4=81$ . So, $x=4$ . Next, find the value of $y$ . We need to determine what power of 2 equals 64. $2^1=2$ , $2^2=4$ , $2^3=8$ , $2^4=16$ , $2^5=32$ , $2^6=64$ . So, $y=6$ . The question asks for the value of $x-y$ , which is $4-6 = -2$ . * B is incorrect. This might result from a calculation error for $x$ or $y$ . * C is incorrect. This might result from a calculation error for $x$ or $y$ . * D is incorrect. This is the value of $y-x$ , not $x-y$ .
The expression $(x^a)^2(x^3)^{-1}$ simplifies to $x^7$ . What is the value of $a$ ?
Answer: 5 First, simplify the left side of the equation using exponent rules. $(x^a)^2 = x^{2a}$ and $(x^3)^{-1} = x^{-3}$ . Multiplying these gives $x^{2a} \cdot x^{-3} = x^{2a-3}$ . We are given that this simplifies to $x^7$ . Therefore, the exponents must be equal: $2a - 3 = 7$ . To solve for $a$ , add 3 to both sides: $2a = 10$ . Then, divide by 2: $a = 5$ . * A is incorrect. If $a=2$ , the exponent would be $2(2)-3=1$ . * B is incorrect. If $a=4$ , the exponent would be $2(4)-3=5$ . * D is incorrect. This could result from an error in solving the equation, such as $2a=10 \rightarrow a=8$ , or from setting up the equation incorrectly, like $2a+3=7$ which gives $2a=4, a=2$ .
What is the value of $(-1)^{100} - (-1)^{101}$ ?
Answer: 2 The value of $-1$ raised to a power depends on whether the exponent is even or odd. For an even exponent, the result is 1. For an odd exponent, the result is -1. In this expression, $(-1)^{100} = 1$ because 100 is even. And $(-1)^{101} = -1$ because 101 is odd. Substituting these values into the expression gives $1 - (-1)$ , which simplifies to $1 + 1 = 2$ . * A is incorrect. This would result from reversing the signs, calculating $-1 - 1 = -2$ . * B is incorrect. This results from either thinking both terms are 1 ( $1-1=0$ ) or both terms are -1 ( $-1 - (-1)=0$ ). * C is incorrect. This could happen if the subtraction is ignored or if the second term is mistakenly evaluated to 0.
If $x \neq 0$ , what is the value of $(5x^3)^2 \cdot (5x^6)^{-1}$ ?
Answer: 5 First, apply the exponents to each factor. The first term becomes $5^2(x^3)^2 = 25x^6$ . The second term becomes $5^{-1}(x^6)^{-1} = \frac{1}{5}x^{-6}$ . Now multiply the results: $(25x^6) \cdot (\frac{1}{5}x^{-6})$ . Group the coefficients and variables: $(25 \cdot \frac{1}{5}) \cdot (x^6 \cdot x^{-6})$ . The coefficients multiply to $\frac{25}{5}=5$ . The variables multiply to $x^{6+(-6)} = x^0$ . Since $x \neq 0$ , $x^0=1$ . The final result is $5 \cdot 1 = 5$ . * A is incorrect. This may result from the misconception that $x^0=0$ . * B is incorrect. This results from an error with the coefficients, such as $5/5=1$ instead of $25/5$ . * D is incorrect. This results from ignoring the second coefficient, $5^{-1}$ , or making a sign error with the exponents, leading to $25x^{12}$ , which doesn't simplify to a constant.
If $y = 2x^{-3}$ and $x = -2$ , what is the value of $y$ ?
Answer: $-\frac{1}{4}$ First, substitute $x = -2$ into the expression for $y$ : $y = 2(-2)^{-3}$ . According to the order of operations, evaluate the exponent first. $(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8}$ . Now, multiply by the coefficient 2: $y = 2 \cdot \left(\frac{1}{-8}\right) = \frac{2}{-8} = -\frac{1}{4}$ . * A is incorrect. This could result from incorrectly calculating $2(-2)^{-3}$ as $2(8)$ or $-2(8)$ . * C is incorrect. This is a sign error, likely from evaluating $(-2)^3$ as 8 instead of -8. * D is incorrect. This may result from multiplying the base by the exponent, as in $2(-2 \cdot -3) = 2(6) = 12$ .
Which of the following values is equal to $\frac{1}{2^{-5}}$ ?
Answer: 32 The expression $\frac{1}{a^{-n}}$ is equivalent to $a^n$ . Therefore, $\frac{1}{2^{-5}}$ is equal to $2^5$ . Calculating $2^5$ gives $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ . * A is incorrect. This comes from incorrectly applying a negative sign to the result. * B is incorrect. This is a result of incorrectly multiplying the base by the exponent ( $2 \cdot 5 = 10$ ) and making a sign error. * C is incorrect. This is the value of $2^{-5}$ , not $\frac{1}{2^{-5}}$ .
What is the value of the expression $\frac{-2^4}{(-4)^2}$ ?
Answer: -1 Evaluate the numerator and the denominator separately. In the numerator, $-2^4$ , the exponent applies only to the 2, not the negative sign. So, $-2^4 = -(2 \cdot 2 \cdot 2 \cdot 2) = -16$ . In the denominator, $(-4)^2$ , the exponent applies to $-4$ . So, $(-4)^2 = (-4)(-4) = 16$ . The fraction is $\frac{-16}{16}$ , which simplifies to -1. * B is incorrect. This could result from an error in calculating the numerator, such as $-2^4 = -8$ . * C is incorrect. This could result from sign errors in both the numerator and denominator. * D is incorrect. This would result from an error in the numerator, such as evaluating $-2^4$ as 16, or an error in the denominator, such as $(-4)^2 = -16$ , leading to $\frac{-16}{-16} = 1$ .
Simplify: $\left(\frac{3^{-2} \cdot 9^2}{27^{-1}}\right)^{-1}$
Answer: $\frac{1}{243}$ First, express everything in terms of base 3: $9^2 = (3^2)^2 = 3^4$ and $27^{-1} = (3^3)^{-1} = 3^{-3}$ . The expression inside the parentheses becomes $\frac{3^{-2} \cdot 3^4}{3^{-3}} = \frac{3^{-2+4}}{3^{-3}} = \frac{3^2}{3^{-3}} = 3^{2-(-3)} = 3^{2+3} = 3^5$ . Therefore, $\left(3^5\right)^{-1} = 3^{-5} = \frac{1}{3^5} = \frac{1}{243}$ . Choice B results from computing $3^{-4}$ due to an error in exponent arithmetic. Choice C results from computing $3^{-3}$ from incorrectly simplifying the inner expression. Choice D results from computing $3^{-2}$ from multiple arithmetic errors.
Which of the following is equivalent to $\left(\frac{x^{-3}}{y^2}\right)^{-2}$ for $x \neq 0, y \neq 0$ ?
Answer: $x^6y^4$ To simplify the expression, apply the outer exponent $-2$ to both the numerator and the denominator: $\frac{(x^{-3})^{-2}}{(y^2)^{-2}}$ . Using the power of a power rule ( $(a^m)^n=a^{mn}$ ), the numerator becomes $x^{(-3)(-2)} = x^6$ and the denominator becomes $y^{(2)(-2)} = y^{-4}$ . The expression is $\frac{x^6}{y^{-4}}$ . Since a negative exponent in the denominator is equivalent to a positive exponent in the numerator, $y^{-4} = \frac{1}{y^4}$ , so $\frac{x^6}{y^{-4}} = x^6y^4$ . * A is incorrect. This may result from incorrectly adding exponents, such as $x^{-3-2}=x^{-5}$ and $y^{2-2}=y^0=1$ . * B is incorrect. This results from correctly applying the outer exponent to the denominator but making a sign error on the numerator's exponent. * C is incorrect. This results from correctly simplifying the numerator but making a sign error on the denominator's exponent.
What is the value of $(10^2 - 10^0)^{-1}$ ?
Answer: $\frac{1}{99}$ Following the order of operations, evaluate the expression inside the parentheses first. $10^2 = 100$ and $10^0 = 1$ . So, the expression inside the parentheses is $100 - 1 = 99$ . The problem then becomes $99^{-1}$ . A negative exponent indicates a reciprocal, so $99^{-1} = \frac{1}{99}$ . * A is incorrect. This results from the common mistake of incorrectly distributing the exponent: $(10^2)^{-1} - (10^0)^{-1} = 10^{-2} - 1 = \frac{1}{100} - 1 = -\frac{99}{100}$ . * B is incorrect. This results from incorrectly evaluating $10^0$ as 0, which gives $(100-0)^{-1} = 100^{-1} = \frac{1}{100}$ . * D is incorrect. This could result from misreading $10^0$ as $10^1$ , which would lead to $(100-10)^{-1} = 90^{-1} = \frac{1}{90}$ .
Which of the following expressions is equivalent to $\frac{(x^4)^3}{x^2}$ for all $x \neq 0$ ?
Answer: $x^{10}$ First, simplify the numerator. According to the power of a power rule, $(a^m)^n = a^{mn}$ , so $(x^4)^3 = x^{4 \cdot 3} = x^{12}$ . Now the expression is $\frac{x^{12}}{x^2}$ . According to the quotient of powers rule, $\frac{a^m}{a^n} = a^{m-n}$ , so $\frac{x^{12}}{x^2} = x^{12-2} = x^{10}$ . * A is incorrect. This results from adding the exponents in the numerator ( $4+3=7$ ) instead of multiplying, giving $\frac{x^7}{x^2} = x^5$ . * B is incorrect. This results from dividing the exponents ( $12 \div 2 = 6$ ) instead of subtracting. * D is incorrect. This results from subtracting the denominator's exponent from the numerator's exponents incorrectly, possibly by adding them, as in $x^{12+2} = x^{14}$ .
Which of the following is equivalent to the expression $\left(\frac{3}{4}\right)^{-2}$ ?
Answer: $\frac{16}{9}$ A negative exponent indicates taking the reciprocal of the base. Therefore, $\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^{2}$ . Squaring the fraction means squaring both the numerator and the denominator: $\frac{4^2}{3^2} = \frac{16}{9}$ . * A is incorrect because the negative exponent does not make the result negative, and the fraction is inverted incorrectly. * B is incorrect because the negative exponent means to take the reciprocal of the base, not to make the result negative. * C is incorrect because it squares the original fraction without taking its reciprocal first, which is what the negative exponent requires.
What is the value of the expression $5 \cdot 10^{-2}$ ?
Answer: 0.05 First, evaluate the exponential term. $10^{-2} = \frac{1}{10^2} = \frac{1}{100}$ , which is equal to the decimal 0.01. Then, multiply this value by 5: $5 \cdot 0.01 = 0.05$ . * A is incorrect. This would be the value of $5 \cdot 10^{-3}$ . * C is incorrect. This results from misinterpreting the negative exponent as positive and adding zeros, as in $5 \cdot 10^2 = 500$ . * D is incorrect. This answer seems to come from an unrelated calculation and is highly implausible.

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