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\(\textbf{1)}\) \( 2x^2 \cdot 3x^3 \)
\(\textbf{2)}\) \( 4x^{-3} \cdot 3x^{4} \)
\(\textbf{3)}\) \( x^3 \cdot 5x \)
\(\textbf{4)}\) \( \displaystyle \frac{12x^4 y^3 z}{3x^2 z^4 x} \)
\(\textbf{5)}\) \( \displaystyle\frac{4x^4 y^{-2} z}{6x^{-1} yz} \)
\(\textbf{6)}\) \( -2^4 \)
\(\textbf{7)}\) \( (-2)^4 \)
\(\textbf{8)}\) \( \displaystyle \left( \frac{2}{5} \right)^3 \)
\(\textbf{9)}\) \( (x^3 )^4 \)
\(\textbf{10)}\) \( (3x^2 y^4 z)^3 \)
\(\textbf{11)}\) \( (2w^4 xz^3 )^5 \)
\(\textbf{12)}\) \( (11x^3 )^2 \)
\(\textbf{13)}\) \( (5zw^2 )^4 \)
\(\textbf{14)}\) \( (3z^2 w^{-5} )^4 \)
\(\textbf{15)}\) \( (2z^3 w^6 )^{-4} \)
\(\textbf{16)}\) \( \displaystyle \frac{5x^6 y^2 z^2}{10x^{-2} z^4 x^3} \)
\(\textbf{17)}\) \( \displaystyle \left(\frac{3x}{y} \right)^2 \)
\(\textbf{18)}\) \( \displaystyle \left(\frac{5x^5 y^{-3} z^2}{6x^{-2} yz} \right)^{-2} \)
\(\textbf{19)}\) \( \displaystyle \frac{3m^{-3}}{\left(3m^{-2}\right)^4} \)
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.
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