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Exponents are a way to express repeated multiplication of the same number. For example, in the expression \[\text{a}^{\text{b}}\], ‘a’ is the base and ‘b’ is the exponent. It means ‘a’ is multiplied by itself ‘b’ times. For example, \[\text{3}^{\text{4}} \] means 3 × 3 × 3 × 3. This is a compact and efficient way to represent large numbers.

Using exponents simplifies operations. They are a fundamental part of algebra and appear frequently in different areas of mathematics. Common operations with exponents include multiplication, division, and raising a power to another power.

Knowing the properties of exponents makes it easier to work with them. Here are some important properties:

• Product of Powers: When multiplying like bases, add the exponents: \[\text{a}^{\text{m}} \times \text{a}^{\text{n}} = \text{a}^{\text{m+n}}\]
• Quotient of Powers: When dividing like bases, subtract the exponents: \[\frac{\text{a}^{\text{m}}}{\text{a}^{\text{n}}} = \text{a}^{\text{m-n}}\]
• Power of a Power: When raising an exponent to another exponent, multiply the exponents: \[\text{(a}^{\text{m}})^{\text{n}} = \text{a}^{\text{m×n}}\]

These properties are essential when solving problems involving exponents, such as expanding and simplifying expressions.

To prove that \[(\text{a}^{\text{b}})^{\text{c}} = \text{a}^{\text{b\cdot c}}\] for positive integers ‘b’ and ‘c’, follow these steps:

First, write down the expression: \[(\text{a}^{\text{b}})^{\text{c}}\]. It means raising ‘a’ to the power of ‘b’, and then raising the result to the power of ‘c’.

Next, recognize that raising a power to another power implies repeated multiplication. If you have \[\text{(a}^{\text{b}})^{\text{c}}\], it is the same as multiplying \[\text{a}^{\text{b}}\] by itself ‘c’ times:
\[\underbrace{\text{a}^{\text{b}} \times \text{a}^{\text{b}} \times \ldots \times \text{a}^{\text{b}}}_{\text{c times}}\]
Use the property of exponents for multiplication (Product of Powers Property): \[\text{a}^{\text{m}} \times \text{a}^{\text{n}} = \text{a}^{\text{m+n}}\]. Thus, we get \[\text{a}^{\text{b}} \times \text{a}^{\text{b}} \times \ldots \times \text{a}^{\text{b}} = \text{a}^{\text{b+b+\ldots+b}}\] (c times).

Finally, simplify the expression. Adding ‘b’ ‘c’ times gives you \[\text{b} \times \text{c}\], so it becomes: \[a^{b+b+...+b}=a^{bc}\]
Thus, we have successfully shown that \[(\text{a}^{\text{b}})^{\text{c}} = \text{a}^{\text{bc}}\], proving the power of a power law for positive integer exponents.