Using Properties of Exponents

When powers with the same base are divided, they can be written as one power where the exponent in the denominator is subtracted from numerator's exponent. For example, $3^6$ divided by $3^4$ can be rewritten using the rule as follows. $$\begin{array}{c}\frac{3^6}{3^4}\Rightarrow 3^{6-4}=3^2\end{array}$$ This rule can be explained by writing the powers as a product. This rule is valid for all values of $m,$ $n$ and when $a\ne 0.$ Note that the rule might lead to non-real solutions when $a<0$ and at least one of the exponents is not an integer.

When a power is raised to another exponent, the powers can be combined into one by multiplying them. For example ${\left(5^2\right)}^3$ can be rewritten with this rule as follows. $$\begin{array}{c}{\left(5^2\right)}^3\Rightarrow 5^{2\cdot 3}=5^6\end{array}$$ This rule can be explained by writing the powers as products. This rule is valid for all values of $m$ and $n$ when $a\ne 0.$ For the rule to be true for $a=0,$ it is necessary that both exponents are greater than $0.$ Note that the rule might lead to non-real solutions when $a<0$ and at least one of the exponents is not an integer.

When the base in an exponential expression is a product raised to some power, the term can be rewritten as a product of two powers. For example, ${\left(2\cdot 5\right)}^3$ can be rewritten using this rule as follows. $$\begin{array}{c}{\left(2\cdot 5\right)}^3\Rightarrow 2^3\cdot 5^3\end{array}$$ This rule can be explained by writing the expression with repeated multiplication. This rule is valid for all values of $m,$ when $a\ne 0$ and $b\ne 0.$ For the rule to be valid when either $a$ or $b$ equals $0$ it is necessary that $m>0.$ Note that the rule might lead to non-real solutions when at least one of the bases is less than $0$ and the exponent is not an integer.

When the base of a power is a fraction, the term can be rewritten as a fraction of two powers. The rule states that the exponent is applied to the numerator and the denominator. For example, ${\left(\frac{6}{5}\right)}^4$ can be rewritten using this rule as follows. $$\begin{array}{c}{\left(\frac{6}{5}\right)}^4\Rightarrow \frac{6^4}{5^4}\end{array}$$ This rule can be explained by writing the power with repeated multiplication. For this rule to be valid it is necessary that $b\ne 0.$ Also, for the rule to be valid when $m\le 0$ it is necessary that $a\ne 0.$ Note that the rule might lead to non-real solutions when at least one of the bases is negative and $m$ is not an integer.

When an exponent of a power is negative, the term can be rewritten as a fraction, with the base in the denominator. Note that the sign of the exponent is changed. For example, $5^{\text{-}3}$ can be rewritten with this rule as follows. $$\begin{array}{c}{5}^{\text{-}3}\Rightarrow \frac{1}{5^3}\end{array}$$ This rule can be explained by writing the exponent $\text{-}3$ as $4-7$ and applying the Quotient of Powers Property. This rule is valid for all values of $n$ and when $a\ne 0.$ Note that the rule might lead to non-real solutions when $a<0$ and $n$ is not an integer.