Exponents are used to show repeated multiplication of a number by itself. For example, 7 × 7 × 7 can be represented as \(7^3\)

##### Laws of exponents

1. For any non-zero term \(a\)

Illustration 1: What is the simplification of \(5^5~×~5^1\)

Solution: \(5^5~×~5^1\)

Illustration 2: What is the simplification of \((-6)^{-4}~×~(-6)^{-7}\)

Solution: (-6)^(-4) = – \(\frac{1}{6^4}\)

\((-6)^{-4}~×~(-6)^{-7}\)

We can state that the law is applicable for negative terms also. Therefore the term m and n can be any integer.

2. \(\frac{a^m}{a^n}\)

Illustration 3: Find the value when \(10^{-5}\)

Solution: \(\frac{10^{-5}}{10^{-3}}\)

3. \((a^m)^n\)

Illustration 4: Express \(8^3\)

Solution: We have, \(2×2×2\)

Therefore, \(8^3\)

4. a^m × b^m =(ab)^m, where a is a non-zero term and m and n are integers.

Illustration 5: Simplify and write the exponential form of: \(\frac{1}{8}~ ×~ 5^{-3}\)

Solution: We know that \(\frac{1}{8}\)

Therefore, \(2^{-3}~×~5^{-3}\)

5. \(\frac{a^m}{b^m}\)

Illustration 6: Simplify the expression and find the value: \(\frac{15^3}{5^3}\)

Solution: \((\frac{15}{5})^3\)

6. \(a^0\)

Illustration 7: What is the value of \(5^0 + 2^2 + 4^0 + 7^1 – 3^1\)

Solution: \(5^0 + 2^2 + 4^0 + 7^1 – 3^1\)

This article covers the basic laws of exponents. For any further query on this topic please install Byju’s the learning app.

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