Exponent Rules - Accuplacer Qas

Which of the following expressions is equivalent to $(x^9)^{1/3} \cdot x^{-2}$ ?
Answer: $x$ First, simplify $(x^9)^{1/3}$ using the power rule for exponents, which means multiplying the exponents: $x^{9 \cdot (1/3)} = x^3$ . Now the expression is $x^3 \cdot x^{-2}$ . Using the product rule, add the exponents: $x^{3 + (-2)} = x^1 = x$ .
If $x^{2n} \cdot x^{3n} = x^{15}$ and $\left(x^n\right)^2 = x^6$ , what is the value of $n$ ?
Answer: $n = 3$ From the first equation using product rule: $x^{2n} \cdot x^{3n} = x^{2n+3n} = x^{5n} = x^{15}$ , so $5n = 15$ and $n = 3$ . From the second equation using power rule: $\left(x^n\right)^2 = x^{2n} = x^6$ , so $2n = 6$ and $n = 3$ . Both equations confirm $n = 3$ . Choice B uses only the second equation incorrectly as $n = \frac{6}{2} = 3$ but assumes this gives $n = 2$ . Choice C incorrectly assumes $n = \frac{15}{3}$ . Choice D simplifies $\frac{15}{5}$ but is written as a fraction rather than the integer 3.
Which of the following expressions is NOT equivalent to $\frac{x^{12}}{y^6}$ ?
Answer: $(x^3y^{-1})^4$ Evaluate each choice to see if it simplifies to $\frac{x^{12}}{y^6}$ .\A) $(x^2y^{-1})^6 = x^{2 \cdot 6}y^{-1 \cdot 6} = x^{12}y^{-6} = \frac{x^{12}}{y^6}$ . This is equivalent.\B) $\left(\frac{x^6}{y^3}\right)^2 = \frac{(x^6)^2}{(y^3)^2} = \frac{x^{12}}{y^6}$ . This is equivalent.\C) $x^6 \cdot (x^2y^{-2})^3 = x^6 \cdot x^{6}y^{-6} = x^{6+6}y^{-6} = x^{12}y^{-6} = \frac{x^{12}}{y^6}$ . This is equivalent.\D) $(x^3y^{-1})^4 = x^{3 \cdot 4}y^{-1 \cdot 4} = x^{12}y^{-4} = \frac{x^{12}}{y^4}$ . This is NOT equivalent.
The expression $\frac{(4 \times 10^7)}{(2 \times 10^{-3})}$ is equivalent to which of the following?
Answer: $2 \times 10^{10}$ To simplify this expression, handle the coefficients and the powers of 10 separately. First, divide the coefficients: $4 \div 2 = 2$ . Next, divide the powers of 10 using the quotient rule for exponents: $\frac{10^7}{10^{-3}} = 10^{7 - (-3)} = 10^{7+3} = 10^{10}$ . Combining the two parts gives the final answer: $2 \times 10^{10}$ .
If $\left(\frac{x^a}{y^b}\right)^c = \frac{x^{12}}{y^{15}}$ and $a = 3$ , $b = 5$ , what is the value of $c$ ?
Answer: $c = 4$ Using the power rule, $\left(\frac{x^a}{y^b}\right)^c = \frac{x^{ac}}{y^{bc}}$ . Substituting $a = 3$ and $b = 5$ : $\frac{x^{3c}}{y^{5c}} = \frac{x^{12}}{y^{15}}$ . This gives us $3c = 12$ and $5c = 15$ . Both equations yield $c = 4$ . Choice B incorrectly uses $c = a$ . Choice C incorrectly uses $c = b$ . Choice D results from incorrectly solving $\frac{12}{3} = \frac{15}{c}$ .
The volume of a rectangular prism is found by multiplying its length, width, and height. If a prism has a length of $3x^2$ , a width of $2x^5$ , and a height of $x$ , which expression represents its volume?
Answer: $6x^8$ To find the volume, multiply the three dimensions: $(3x^2)(2x^5)(x)$ . First, multiply the numerical coefficients: $3 \cdot 2 = 6$ . Next, multiply the variable terms using the product rule for exponents, remembering that $x = x^1$ : $x^2 \cdot x^5 \cdot x^1 = x^{2+5+1} = x^8$ . Combining the coefficient and the variable gives the volume as $6x^8$ .
The expression $\frac{(2x^5y)^2}{x^3y^3}$ simplifies to the form $ax^by^c$ . What is the value of $b$ ?
Answer: 7 First, simplify the numerator by applying the exponent 2 to each factor inside the parentheses: $(2x^5y)^2 = 2^2(x^5)^2y^2 = 4x^{10}y^2$ . The expression becomes $\frac{4x^{10}y^2}{x^3y^3}$ . Now, simplify the fraction. The coefficient is 4. For the variable $x$ , use the quotient rule: $x^{10-3} = x^7$ . For the variable $y$ , use the quotient rule: $y^{2-3} = y^{-1}$ . The simplified expression is $4x^7y^{-1}$ . In the form $ax^by^c$ , we have $a=4$ , $b=7$ , and $c=-1$ . The question asks for the value of $b$ , which is 7.
The number of bacteria in a lab sample is modeled by the expression $500 \cdot (2^h)$ , where $h$ is the number of hours. Which expression represents the number of bacteria after $3h$ hours?
Answer: $500 \cdot (8^h)$ To find the number of bacteria after $3h$ hours, substitute $3h$ for $h$ in the given expression: $500 \cdot (2^{3h})$ . Using the rules of exponents, $2^{3h}$ can be rewritten as $(2^3)^h$ . Since $2^3 = 8$ , the expression is equivalent to $500 \cdot (8^h)$ .
If $p = (2^3)^2$ and $q = 2^{(3^2)}$ , what is the relationship between $p$ and $q$ ?
Answer: $p < q$ First, evaluate $p$ . Using the power of a power rule, $p = (2^3)^2 = 2^{3 \cdot 2} = 2^6$ . We know $2^6 = 64$ . Next, evaluate $q$ . The exponent is $3^2$ , which must be calculated first: $3^2 = 9$ . So, $q = 2^9$ . We know $2^9 = 512$ . Since $64 < 512$ , it follows that $p < q$ .
What is the value of the expression $\frac{2^8 \cdot 2^{-3}}{2^2}$ ?
Answer: 8 First, simplify the numerator using the product rule for exponents ( $a^x \cdot a^y = a^{x+y}$ ): $2^8 \cdot 2^{-3} = 2^{8+(-3)} = 2^5$ . Now the expression is $\frac{2^5}{2^2}$ . Next, use the quotient rule for exponents ( $\frac{a^x}{a^y} = a^{x-y}$ ): $\frac{2^5}{2^2} = 2^{5-2} = 2^3$ . Finally, calculate the value: $2^3 = 2 \cdot 2 \cdot 2 = 8$ .
The side length of a square is $5x^3y^2$ . What is the area of the square?
Answer: $25x^6y^4$ The area of a square is given by the formula $A = s^2$ , where $s$ is the side length. In this case, $s = 5x^3y^2$ . So, the area is $(5x^3y^2)^2$ . To simplify, apply the exponent 2 to each factor inside the parentheses: $5^2(x^3)^2(y^2)^2$ . This simplifies to $25x^{3 \cdot 2}y^{2 \cdot 2} = 25x^6y^4$ .
Which of the following is equivalent to $(3x^{-2})^{-2} \cdot x^3$ ?
Answer: $\frac{x^7}{9}$ First, apply the power rule to $(3x^{-2})^{-2}$ . The exponent -2 applies to both the 3 and the $x^{-2}$ . This gives $3^{-2} \cdot (x^{-2})^{-2} = 3^{-2} \cdot x^{4}$ . Since $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$ , this part simplifies to $\frac{1}{9}x^4$ . Now, multiply this by the second part of the expression: $(\frac{1}{9}x^4) \cdot x^3$ . Using the product rule, add the exponents of x: $x^4 \cdot x^3 = x^{4+3} = x^7$ . The final result is $\frac{1}{9}x^7$ , or $\frac{x^7}{9}$ .
Which expression is equivalent to $\frac{(3^2)^x \cdot 3^{2x-1}}{3^{x+2}}$ ?
Answer: $3^{3x-3}$ First, apply the power rule: $(3^2)^x = 3^{2x}$ . The expression becomes $\frac{3^{2x} \cdot 3^{2x-1}}{3^{x+2}}$ . Using the product rule in the numerator: $3^{2x} \cdot 3^{2x-1} = 3^{2x + 2x - 1} = 3^{4x-1}$ . Then using the quotient rule: $\frac{3^{4x-1}}{3^{x+2}} = 3^{(4x-1)-(x+2)} = 3^{4x-1-x-2} = 3^{3x-3}$ . Choice B omits the $-2$ from the quotient rule. Choice C incorrectly simplifies the numerator. Choice D adds exponents instead of subtracting in the quotient rule.
Which of the following is equivalent to $\frac{(4p^5q^0)^2}{2p^3}$ ?
Answer: $8p^7$ First, simplify the term $q^0$ , which is equal to 1. The expression becomes $\frac{(4p^5)^2}{2p^3}$ . Next, apply the exponent 2 to the terms in the numerator: $(4p^5)^2 = 4^2(p^5)^2 = 16p^{10}$ . The expression is now $\frac{16p^{10}}{2p^3}$ . Finally, simplify the fraction. Divide the coefficients: $16 \div 2 = 8$ . Use the quotient rule for the variable: $p^{10} \div p^3 = p^{10-3} = p^7$ . The final simplified expression is $8p^7$ .
Simplify the expression $\frac{m^{-5}n^6}{m^2n^{-3}}$ .
Answer: $\frac{n^9}{m^7}$ To simplify the expression, use the quotient rule for exponents, which states $\frac{a^x}{a^y} = a^{x-y}$ . For the variable $m$ , the new exponent is $-5 - 2 = -7$ . For the variable $n$ , the new exponent is $6 - (-3) = 6 + 3 = 9$ . This gives the expression $m^{-7}n^9$ . To express this with positive exponents, move the term with the negative exponent to the denominator, resulting in $\frac{n^9}{m^7}$ .
Simplify the expression $\frac{(k^{3n+1})^2}{k^{2n}}$ .
Answer: $k^{4n+2}$ First, simplify the numerator using the power rule, $(a^m)^n = a^{mn}$ . We must distribute the 2 to both terms in the exponent: $(k^{3n+1})^2 = k^{2(3n+1)} = k^{6n+2}$ . The expression now becomes $\frac{k^{6n+2}}{k^{2n}}$ . Now, use the quotient rule, $\frac{a^m}{a^n} = a^{m-n}$ : $k^{(6n+2) - 2n} = k^{4n+2}$ .
If $a^{2x} \cdot a^{3x-1} = a^{13}$ , what is the value of $x$ ?
Answer: $x = \frac{14}{5}$ Using the product rule for exponents, $a^{2x} \cdot a^{3x-1} = a^{2x + 3x - 1} = a^{5x-1}$ . Setting this equal to $a^{13}$ : $5x - 1 = 13$ , so $5x = 14$ , therefore $x = \frac{14}{5}$ . Choice B incorrectly sets $5x = 13$ . Choice C assumes $2x + 3x = 13$ without considering the $-1$ . Choice D results from the error $5x - 1 = 12$ .
Simplify the expression $x^{-5} \cdot \frac{x^8}{x^{-2}}$ .
Answer: $x^5$ First, simplify the fraction $\frac{x^8}{x^{-2}}$ using the quotient rule for exponents: $x^{8 - (-2)} = x^{8+2} = x^{10}$ . Now the expression is $x^{-5} \cdot x^{10}$ . Next, use the product rule for exponents to combine these terms: $x^{-5+10} = x^5$ .
The expression $\frac{2x^0y^5}{(2y)^3}$ is equivalent to which of the following?
Answer: $\frac{y^2}{4}$ First, simplify terms with zero exponents: $x^0 = 1$ . The expression becomes $\frac{2(1)y^5}{(2y)^3} = \frac{2y^5}{(2y)^3}$ . Next, simplify the denominator by applying the exponent 3 to both factors: $(2y)^3 = 2^3y^3 = 8y^3$ . The expression is now $\frac{2y^5}{8y^3}$ . Simplify the coefficients: $\frac{2}{8} = \frac{1}{4}$ . Simplify the variables using the quotient rule: $\frac{y^5}{y^3} = y^{5-3} = y^2$ . Combining these results gives $\frac{1}{4}y^2$ , or $\frac{y^2}{4}$ .
The expression $(a^{-2}b^4)(a^3b^{-1})^{-2}$ is equivalent to which of the following?
Answer: $\frac{b^6}{a^8}$ First, apply the power rule to the second part of the expression: $(a^3b^{-1})^{-2} = a^{3 \cdot -2}b^{-1 \cdot -2} = a^{-6}b^2$ . Now, multiply this result by the first part of the expression using the product rule (add exponents): $(a^{-2}b^4)(a^{-6}b^2) = a^{-2 + (-6)}b^{4 + 2} = a^{-8}b^6$ . Finally, write the expression with positive exponents: $a^{-8}b^6 = \frac{b^6}{a^8}$ .

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