\(\textbf{1)}\) \( 2x^2 \cdot 3x^3 \)
The answer is \( 6x^5 \)
\(\textbf{2)}\) \( 4x^{-3} \cdot 3x^{4} \)
The answer is \( 12x \)
\(\textbf{3)}\) \( x^3 \cdot 5x \)
The answer is \( 5x^4 \)
\(\textbf{4)}\) \( \displaystyle \frac{12x^4 y^3 z}{3x^2 z^4 x} \)
The answer is \( 4xy^3 z^{-3} \) or \( \displaystyle \frac{4xy^3}{z^3} \)
\(\textbf{5)}\) \( \displaystyle\frac{4x^4 y^{-2} z}{6x^{-1} yz} \)
The answer is \( \displaystyle\frac{2x^5}{3y^3} \) or \( \displaystyle\frac{2}{3} x^5 y^{-3} \)
\(\textbf{6)}\) \( -2^4 \)
The answer is \( -16 \)
\(\textbf{7)}\) \( (-2)^4 \)
The answer is \( 16 \)
\(\textbf{8)}\) \( \displaystyle \left( \frac{2}{5} \right)^3 \)
The answer is \( \displaystyle\frac{8}{125} \)
\(\textbf{9)}\) \( (x^3 )^4 \)
The answer is \( x^{12} \)
\(\textbf{10)}\) \( (3x^2 y^4 z)^3 \)
The answer is \( 27x^6 y^{12} z^3 \)
\(\textbf{11)}\) \( (2w^4 xz^3 )^5 \)
The answer is \( 32w^{20} x^5 z^{15} \)
\(\textbf{12)}\) \( (11x^3 )^2 \)
The answer is \( 121x^6 \)
\(\textbf{13)}\) \( (5zw^2 )^4 \)
The answer is \( 625z^4 w^8 \)
\(\textbf{14)}\) \( (3z^2 w^{-5} )^4 \)
The answer is \( 81z^8 w^{-20} \) or \(\displaystyle\frac{81z^8}{w^{20}} \)
\(\textbf{15)}\) \( (2z^3 w^6 )^{-4} \)
The answer is \( \displaystyle\frac{1}{16} z^{-12} w^{-24} \) or \( \displaystyle\frac{1}{16z^{12} w^{24}}\)
\(\textbf{16)}\) \( \displaystyle \frac{5x^6 y^2 z^2}{10x^{-2} z^4 x^3} \)
The answer is \( \displaystyle \frac{x^5y^2}{2z^2} \)
\(\textbf{17)}\) \( \displaystyle \left(\frac{3x}{y} \right)^2 \)
The answer is \( \displaystyle\frac{9x^2}{y^2} \)
\(\textbf{18)}\) \( \displaystyle \left(\frac{5x^5 y^{-3} z^2}{6x^{-2} yz} \right)^{-2} \)
The answer is \( \displaystyle\frac{36y^8}{25x^{14} z^2} \)
\(\textbf{19)}\) \( \displaystyle \frac{3m^{-3}}{\left(3m^{-2}\right)^4} \)
The answer is \( \displaystyle \frac{m^5}{27} \)
\(\textbf{20)}\) Evaluate \(3^{-3}\)
\(\frac{1}{27}\)
\(\,\,\,\text{Step 1: Recall the rule for negative exponents.}\)
\(\,\,\,\,\,\,a^{-n} = \frac{1}{a^n}\)
\(\,\,\,\text{Step 2: Apply this rule to } 3^{-3}.\)
\(\,\,\,\,\,\,3^{-3} = \frac{1}{3^3} = \frac{1}{27}\)
\(\,\,\,\text{The solution is } \frac{1}{27}.\)
\(\textbf{21)}\) Evaluate \((-2)^4\)
\(16\)
\(\,\,\,\text{Step 1: Use the rule for exponents.}\)
\(\,\,\,\,\,\,(-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2)\)
\(\,\,\,\text{Step 2: Simplify step by step.}\)
\(\,\,\,\,\,\,(-2) \cdot (-2) = 4\)
\(\,\,\,\,\,\,4 \cdot (-2) = -8\)
\(\,\,\,\,\,\,-8 \cdot (-2) = 16\)
\(\,\,\,\text{The solution is } 16.\)
\(\textbf{22)}\) Simplify \((\frac{1}{3})^{-2}\)
\(9\)
\(\,\,\,\text{Step 1: Use the rule for negative exponents.}\)
\(\,\,\,\,\,\,a^{-n} = \frac{1}{a^n}\)
\(\,\,\,\text{Step 2: Simplify } (\frac{1}{3})^{-2}.\)
\(\,\,\,\,\,\,(\frac{1}{3})^{-2} = \frac{1}{(\frac{1}{3})^2} = \frac{1}{\frac{1}{9}}\)
\(\,\,\,\text{Step 3: Simplify further.}\)
\(\,\,\,\,\,\,\frac{1}{\frac{1}{9}} = 9\) \(\,\,\,\text{The solution is } 9.\)
\(\textbf{23)}\) Evaluate \((-4)^{-2}\)
\(\frac{1}{16}\)
\(\,\,\,\text{Step 1: Recall the rule for negative exponents.}\)
\(\,\,\,\,\,\,a^{-n} = \frac{1}{a^n}\)
\(\,\,\,\text{Step 2: Apply this rule to } (-4)^{-2}.\)
\(\,\,\,\,\,\,(-4)^{-2} = \frac{1}{(-4)^2}\) \(\,\,\,\,\,\,\frac{1}{(-4)^2} = \frac{1}{16}\)
\(\,\,\,\text{The solution is } \frac{1}{16}.\)
\(\textbf{24)}\) Simplify \((2^3)^2\)
\(64\)
\(\,\,\,\text{Step 1: Use the power rule } (a^m)^n = a^{m \cdot n}.\)
\(\,\,\,\,\,\,(2^3)^2 = 2^{3 \cdot 2}\)
\(\,\,\,\,\,\,2^{3 \cdot 2} = 2^6\)
\(\,\,\,\text{Step 2: Simplify further.}\)
\(\,\,\,\,\,\,2^6 = 64\)
\(\,\,\,\text{The solution is } 64.\)
\(\textbf{25)}\) Evaluate \((9^{\frac{1}{2}})^2\)
\(9\)
\(\,\,\,\text{Step 1: Use the power rule } (a^m)^n = a^{m \cdot n}.\)
\(\,\,\,\,\,\,(9^{\frac{1}{2}})^2 = 9^{\frac{1}{2} \cdot 2}\)
\(\,\,\,\,\,\,9^{\frac{1}{2} \cdot 2} = 9^1\)
\(\,\,\,\text{Step 2: Simplify.}\)
\(\,\,\,\,\,\,9^1 = 9\)
\(\,\,\,\text{The solution is } 9.\)
\(\textbf{26)}\) Simplify \(\frac{3^{-2}}{3^3}\)
\(\frac{1}{243}\)
\(\,\,\,\text{Step 1: Use the quotient rule } \frac{a^m}{a^n} = a^{m-n}.\)
\(\,\,\,\,\,\,\frac{3^{-2}}{3^3} = 3^{-2-3}\)
\(\,\,\,\,\,\,3^{-5} = \frac{1}{3^5}\)
\(\,\,\,\text{Step 2: Simplify further.}\)
\(\,\,\,\,\,\,\frac{1}{3^5} = \frac{1}{243}\)
\(\,\,\,\text{The solution is } \frac{1}{243}.\)
\(\textbf{27)}\) Evaluate \((-5)^0\)
\(1\)
\(\,\,\,\text{Step 1: Recall the zero exponent rule.}\)
\(\,\,\,\,\,\,a^0 = 1 \text{ for any non-zero value of } a.\)
\(\,\,\,\text{Step 2: Apply the rule.}\)
\(\,\,\,\,\,\,(-5)^0 = 1\)
\(\,\,\,\text{The solution is } 1.\)
\(\textbf{28)}\) Simplify \((\frac{4}{5})^{-3}\)
\(\frac{125}{64}\)
\(\,\,\,\text{Step 1: Use the rule for negative exponents.}\)
\(\,\,\,\,\,\,(\frac{a}{b})^{-n} = (\frac{b}{a})^n\)
\(\,\,\,\text{Step 2: Apply the rule.}\)
\(\,\,\,\,\,\,(\frac{4}{5})^{-3} = (\frac{5}{4})^3\)
\(\,\,\,\text{Step 3: Expand.}\)
\(\,\,\,\,\,\,(\frac{5}{4})^3 = \frac{5^3}{4^3} = \frac{125}{64}\)
\(\,\,\,\text{The solution is } \frac{125}{64}.\)
\(\textbf{29)}\) Evaluate \(2^{-3} \cdot 2^4\)
\(2\)
\(\,\,\,\text{Step 1: Use the product rule } a^m \cdot a^n = a^{m+n}.\)
\(\,\,\,\,\,\,2^{-3} \cdot 2^4 = 2^{-3+4}\)
\(\,\,\,\,\,\,2^1 = 2\)
\(\,\,\,\text{The solution is } 2.\)
\(\textbf{30)}\) Simplify \((5^2 \cdot 5^{-3})\)
\(\frac{1}{5}\)
\(\,\,\,\text{Step 1: Use the product rule } a^m \cdot a^n = a^{m+n}.\)
\(\,\,\,\,\,\,5^2 \cdot 5^{-3} = 5^{2+(-3)}\)
\(\,\,\,\,\,\,5^{-1} = \frac{1}{5}\)
\(\,\,\,\text{The solution is } \frac{1}{5}.\)
\(\textbf{31)}\) Evaluate \((6^{-2} \cdot 6^4)\)
\(36\)
\(\,\,\,\text{Step 1: Use the product rule } a^m \cdot a^n = a^{m+n}.\)
\(\,\,\,\,\,\,6^{-2} \cdot 6^4 = 6^{-2+4}\)
\(\,\,\,\,\,\,6^2 = 36\)
\(\,\,\,\text{The solution is } 36.\)
\(\textbf{32)}\) Simplify \((2^3)^4\)
\(4096\)
\(\,\,\,\text{Step 1: Use the power rule } (a^m)^n = a^{m \cdot n}.\)
\(\,\,\,\,\,\,(2^3)^4 = 2^{3 \cdot 4}\)
\(\,\,\,\,\,\,2^{12} = 4096\)
\(\,\,\,\text{The solution is } 4096.\)
\(\textbf{33)}\) Evaluate \(\frac{4^{-2}}{2^{-3}}\)
\(\frac{1}{64}\)
\(\,\,\,\text{Step 1: Simplify each base.}\)
\(\,\,\,\,\,\,4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
\(\,\,\,\,\,\,2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
\(\,\,\,\text{Step 2: Divide the two fractions.}\)
\(\,\,\,\,\,\,\frac{\frac{1}{16}}{\frac{1}{8}} = \frac{1}{16} \cdot 8 = \frac{1}{2} \cdot 8 = \frac{1}{64}\)
\(\,\,\,\text{The solution is } \frac{1}{64}.\)
\(\textbf{34)}\) Simplify \((\frac{2}{3})^3 \cdot (\frac{2}{3})^{-1}\)
\((\frac{2}{3})^2\)
\(\,\,\,\text{Step 1: Use the product rule } a^m \cdot a^n = a^{m+n}.\)
\(\,\,\,\,\,\,(\frac{2}{3})^3 \cdot (\frac{2}{3})^{-1} = (\frac{2}{3})^{3+(-1)}\)
\(\,\,\,\,\,\,(\frac{2}{3})^{2}\)
\(\,\,\,\text{The solution is } (\frac{2}{3})^2.\)
\(\textbf{35)}\) Simplify \((7^{-3})^{-2}\)
\(7^6\)
\(\,\,\,\text{Step 1: Use the power rule } (a^m)^n = a^{m \cdot n}.\)
\(\,\,\,\,\,\,(7^{-3})^{-2} = 7^{-3 \cdot -2}\)
\(\,\,\,\,\,\,7^6\)
\(\,\,\,\text{The solution is } 7^6.\)
\(\textbf{36)}\) Evaluate \((\frac{1}{2})^{-4}\)
\(16\)
\(\,\,\,\text{Step 1: Use the rule for negative exponents.}\)
\(\,\,\,\,\,\,(\frac{1}{2})^{-4} = (\frac{2}{1})^4\)
\(\,\,\,\text{Step 2: Simplify.}\)
\(\,\,\,\,\,\,2^4 = 16\)
\(\,\,\,\text{The solution is } 16.\)
\(\textbf{37)}\) Simplify \((x^3)^2 \cdot x^{-4}\)
\(x^2\)
\(\,\,\,\text{Step 1: Use the power rule } (a^m)^n = a^{m \cdot n}.\)
\(\,\,\,\,\,\,(x^3)^2 \cdot x^{-4} = x^{3 \cdot 2} \cdot x^{-4}\)
\(\,\,\,\text{Step 2: Use the product rule } a^m \cdot a^n = a^{m+n}.\)
\(\,\,\,\,\,\,x^6 \cdot x^{-4} = x^{6-4} = x^2\)
\(\,\,\,\text{The solution is } x^2.\)
\(\textbf{38)}\) Evaluate \((4^{-1})^2\)
\(\frac{1}{16}\)
\(\,\,\,\text{Step 1: Use the power rule.}\)
\(\,\,\,\,\,\,(4^{-1})^2 = 4^{-2}\)
\(\,\,\,\text{Step 2: Simplify.}\)
\(\,\,\,\,\,\,4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
\(\,\,\,\text{The solution is } \frac{1}{16}.\)
\(\textbf{39)}\) Simplify \(\frac{x^5}{x^2}\)
\(x^3\)
\(\,\,\,\text{Step 1: Use the quotient rule } \frac{a^m}{a^n} = a^{m-n}.\)
\(\,\,\,\,\,\,\frac{x^5}{x^2} = x^{5-2}\) \(\,\,\,\,\,\,x^3\)
\(\,\,\,\text{The solution is } x^3.\)
\(\textbf{40)}\) Simplify \((3x^2y^3)^2\)
\(9x^4y^6\)
\(\,\,\,\text{Step 1: Use the power rule } (ab)^n = a^n \cdot b^n.\)
\(\,\,\,\,\,\,(3x^2y^3)^2 = 3^2 \cdot (x^2)^2 \cdot (y^3)^2\)
\(\,\,\,\text{Step 2: Simplify each term.}\)
\(\,\,\,\,\,\,3^2 = 9, \, (x^2)^2 = x^{2 \cdot 2} = x^4, \, (y^3)^2 = y^{3 \cdot 2} = y^6\)
\(\,\,\,\,\,\,9x^4y^6\)
\(\,\,\,\text{The solution is } 9x^4y^6.\)
\(\textbf{41)}\) Evaluate \(\frac{2^5 \cdot 3^{-1}}{6^2}\)
\(\frac{8}{27}\)
\(\,\,\,\text{Step 1: Simplify the numerator and denominator.}\)
\(\,\,\,\,\,\,2^5 = 32, \, 3^{-1} = \frac{1}{3}, \, 6^2 = 36\)
\(\,\,\,\,\,\,\frac{2^5 \cdot 3^{-1}}{6^2} \)
\(\,\,\,\,\,\,\frac{32}{3 \cdot 36} \)
\(\,\,\,\,\,\,\frac{32}{108} \)
\(\,\,\,\,\,\,\frac{8}{27} \)
\(\,\,\,\text{The solution is } \frac{8}{27}.\)
\(\textbf{42)}\) Simplify \(\frac{a^{-3}b^4}{a^2b^{-2}}\)
\(\frac{b^6}{a^5}\)
\(\,\,\,\text{Step 1: Simplify each variable separately using the quotient rule.}\)
\(\,\,\,\,\,\,\frac{a^{-3}}{a^2} = a^{-3-2} = a^{-5} = \frac{1}{a^5}\)
\(\,\,\,\,\,\,\frac{b^4}{b^{-2}} = b^{4-(-2)} = b^{4+2} = b^6\)
\(\,\,\,\text{Step 2: Combine the results.}\)
\(\,\,\,\,\,\,\frac{a^{-3}b^4}{a^2b^{-2}} = \frac{b^6}{a^5}\)
\(\,\,\,\text{The solution is } \frac{b^6}{a^5}.\)
\(\textbf{43)}\) Simplify \((\frac{x^3}{y^{-2}})^2\)
\(\frac{x^6}{y^{-4}}\)
\(\,\,\,\text{Step 1: Use the power rule } (a^m)^n = a^{m \cdot n}.\)
\(\,\,\,\,\,\,(\frac{x^3}{y^{-2}})^2 = \frac{(x^3)^2}{(y^{-2})^2}\)
\(\,\,\,\text{Step 2: Simplify each term.}\)
\(\,\,\,\,\,\,(x^3)^2 = x^{3 \cdot 2} = x^6\)
\(\,\,\,\,\,\,(y^{-2})^2 = y^{-2 \cdot 2} = y^{-4}\)
\(\,\,\,\,\,\,\frac{x^6}{y^{-4}}\)
\(\,\,\,\text{The solution is } \frac{x^6}{y^{-4}}.\)
\(\textbf{44)}\) Simplify \((2^4 \cdot 2^{-3}) \div 2^2\)
\(2^{-1}\)
\(\,\,\,\text{Step 1: Simplify the numerator using the product rule.}\)
\(\,\,\,\,\,\,2^4 \cdot 2^{-3} = 2^{4-3} = 2^1\)
\(\,\,\,\text{Step 2: Simplify the division using the quotient rule.}\)
\(\,\,\,\,\,\,2^1 \div 2^2 = 2^{1-2} = 2^{-1}\) \(\,\,\,\text{The solution is } 2^{-1}.\)
\(\textbf{45)}\) Simplify \(\frac{3x^{-2}y^4}{9x^3y^{-5}}\)
\(\frac{y^9}{3x^5}\)
\(\,\,\,\text{Step 1: Simplify the coefficients.}\)
\(\,\,\,\,\,\,\frac{3}{9} = \frac{1}{3}\)
\(\,\,\,\text{Step 2: Use the quotient rule on } x \text{ and } y.\)
\(\,\,\,\,\,\,\frac{x^{-2}}{x^3} = x^{-2-3} = x^{-5}\)
\(\,\,\,\,\,\,\frac{y^4}{y^{-5}} = y^{4-(-5)} = y^{4+5} = y^9\)
\(\,\,\,\text{Step 3: Combine everything.}\)
\(\,\,\,\,\,\,\frac{3x^{-2}y^4}{9x^3y^{-5}} = \frac{y^9}{3x^5}\)
\(\,\,\,\text{The solution is } \frac{y^9}{3x^5}.\)
\(\textbf{46)}\) Evaluate \(\displaystyle \left( \frac{-3x^{4n}}{x^{2n} y^5} \right)^3\)
\(\frac{-27x^{6n}}{y^{15}}\)
\(\,\,\,\,\,\,\text{Step 1: Simplify the fraction inside the parentheses.}\)
\(\,\,\,\,\,\,x^{4n} \div x^{2n} = x^{4n – 2n} = x^{2n}\)
\(\,\,\,\,\,\,\text{The expression becomes } \frac{-3x^{2n}}{y^5}.\)
\(\,\,\,\,\,\,\text{Step 2: Apply the cube to both the numerator and denominator.}\)
\(\,\,\,\,\,\,\left( \frac{-3x^{2n}}{y^5} \right)^3 = \frac{(-3)^3 (x^{2n})^3}{(y^5)^3} = \frac{-27x^{6n}}{y^{15}}\)
\(\,\,\,\,\,\,\text{The solution is } \frac{-27x^{6n}}{y^{15}}.\)
\(\textbf{47)}\) Evaluate \(\displaystyle \left( \frac{5x^{n+1}}{2x^n y^4} \right)^2\)
\(\frac{25x^2}{4y^8}\)
\(\,\,\,\,\,\,\text{Step 1: Simplify the fraction inside the parentheses.}\)
\(\,\,\,\,\,\,x^{n+1} \div x^n = x^{(n+1) – n} = x^1 = x\)
\(\,\,\,\,\,\,\text{The expression becomes } \frac{5x}{2y^4}.\)
\(\,\,\,\,\,\,\text{Step 2: Square both the numerator and denominator.}\)
\(\,\,\,\,\,\,\left( \frac{5x}{2y^4} \right)^2 = \frac{(5x)^2}{(2y^4)^2} = \frac{25x^2}{4y^8}\)
\(\,\,\,\,\,\,\text{The solution is } \frac{25x^2}{4y^8}.\)
\(\textbf{48)}\) Evaluate \(\displaystyle \left( \frac{-4x^{5m}}{8x^{2m} y^2} \right)^4\)
\(\frac{x^{12m}}{16y^8}\)
\(\,\,\,\,\,\,\text{Step 1: Simplify the fraction inside the parentheses.}\)
\(\,\,\,\,\,\,x^{5m} \div x^{2m} = x^{5m – 2m} = x^{3m}\)
\(\,\,\,\,\,\,\text{The expression becomes } \frac{-4x^{3m}}{8y^2}.\)
\(\,\,\,\,\,\,\text{Step 2: Simplify the coefficient and apply the exponent.}\)
\(\,\,\,\,\,\,\frac{-4}{8} = -\frac{1}{2}, \quad \left( -\frac{1}{2} \right)^4 = \frac{1}{16}\)
\(\,\,\,\,\,\,\text{Step 3: Apply the fourth power to the variables.}\)
\(\,\,\,\,\,\,\left( x^{3m} \right)^4 = x^{12m}, \quad \left( y^2 \right)^4 = y^8\)
\(\,\,\,\,\,\,\text{The solution is } \frac{x^{12m}}{16y^8}.\)
In Summary
The properties of exponents are the rules and principles that govern the behavior of exponents, or powers, in algebraic expressions. The properties of exponents are typically introduced in a high school or college algebra class.
A common mistake when working with the properties of exponents is to forget the rules for multiplying or dividing powers with the same base. It is important to remember that when two powers with the same base are multiplied together, the exponents are added, and when two powers with the same base are divided, the exponents are subtracted. Another common mistake is to forget the rule for raising a power to a power, which states that the exponents are multiplied.
Exponents were originally used by medieval scribes to represent repeated multiplication in a more compact form.
