Law of exponents — lesson. Mathematics CBSE, Class 7.

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Basic properties of Exponents.

\begin{array}{l} a^{n} \cdot a^{m} = a^{n + m}; \\ a^{n} : a^{m} = a^{n - m}, n > m, a \neq 0; \\ {(a^{n})}^{m} = a^{n \cdot m} \end{array}

Where n and \(m\) are integers.

In practice, several properties are often applied simultaneously.

Example:

Calculate

\frac{{(5^{4} \cdot 5)}^{3}}{5^{7} \cdot 5^{6}}

Perform actions in the numerator:

5^{4} \cdot 5 = 5^{4} \cdot 5^{1} = 5^{4 + 1} = 5^{5}

— applied property

a^{n} \cdot a^{m} = a^{n + m}

{(5^{5})}^{3} = 5^{5 \cdot 3} = 5^{15}

— applied property

{(a^{n})}^{m} = a^{n \cdot m}

Perform the multiplication in the denominator:

5^{7} \cdot 5^{6} = 5^{7 + 6} = 5^{13}

— applied property

a^{n} \cdot a^{m} = a^{n + m}

Replace the fraction line with division:

5^{15} : 5^{13} = 5^{15 - 13} = 5^{2}

— applied property

a^{n} : a^{m} = a^{n - m}

5^{2} = 25

Answer: \(25\).

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7. Law of exponents