Law of exponents — lesson. Mathematics CBSE, Class 8.

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In the previous class, we have learned about the laws of exponent.

Let's have a quick recall of the laws of exponent.

The laws are:

1. Product law

According to the product law, the exponents can be added when multiplying two powers with the same base.

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

, where \(a ≠ 0\) and \(a\), \(m\), \(n\) are integers.

Example:

\( 10 ^2 × 10 ^5\)

Here, the base \(10\) is same for powers. So we can add the exponents using the product law.

10^{2} \times 10^{5} = 10^{2 + 5} = 10^{7}

2. Quotient law

The quotient law states that we can divide two powers with the same base by subtracting the exponents.

a^{n} : a^{m} = \frac{a^{n}}{a^{m}} = a^{n - m}, n > m, a \neq 0;

, where, \(a\), \(m\), \(n\) are integers.

Example:

\frac{2^{15}}{2^{13}} = 2^{15 - 13} = 2^{2}

\frac{10^{8}}{10^{5}} = 10^{8 - 5} = 10^{3}

3. Power law

The power law states that when a number is raised to a power of another power, we need to multiply the powers or exponents.

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

, where \(a ≠ 0\) and \(a\), \(m\), \(n\) are integers.

Example:

\begin{array}{l} {(2^{5})}^{3} = 2^{5 \times 3} = 2^{15} \\ {(10^{4})}^{2} = 10^{4 + 2} = 10^{6} \end{array}

Powers with Negative Exponent

A number with negative exponent is equal to the reciprocal of the number with positive exponent.

That is,

a^{- n} = \frac{1}{a^{n}}

, here \(n\) is an integer.

If the negative number \((-1)\) raised to the negative odd power \((\)\(-1\)\(^\text{odd power}\)\()\), then the resultant value is negative \((-1)\).

If the negative number \((-1)\) raised to the negative even power \((\)\(-1\)\(^\text{even power}\)\()\), then the resultant value is positive \((1)\).

Example:

5^{- 2} = \frac{1}{5^{2}} = \frac{1}{5 \times 5} = \frac{1}{25}

{- 10}^{- 3} = - \frac{1}{10^{3}} = - \frac{1}{10 \times 10 \times 10} = \frac{1}{1000}

Click here to see more examples from the previous class.

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2. Law of exponents

Theory: