10.3. Laws of Exponents

We have learnt that for any non-zero integer a, am × an = am + n, where m and n are natural numbers. Does this law also hold if the exponents are negative? Let us explore.

In this introduction, we establish a foundational property of exponents: when multiplying two numbers with the same base, you can simply add their exponents. This is represented by the formula: $a^m \times a^n = a^{m+n}$. This chunk raises a question whether the same applies when dealing with negative exponents, which leads us to explore further.

Think of exponents like a recipe. Just as you combine quantities of ingredients to get the final dish, the exponent allows us to combine quantities of the same base number.

(i) We know that 2 – 3 = a−m for any non-zero integer a. Therefore, 2−3 ×2−2 = × = = = 2 – 5.

Here, we explore how the law works with negative exponents. If we take $2^{-3}$ and $2^{-2}$ and multiply them, we use the rule of adding the exponents: $2^{-3} \times 2^{-2} = 2^{-3 + (-2)} = 2^{-5}$. This shows that the law holds true even when the exponents are negative.

Imagine if you’re playing a game where gaining points is like using positive exponents, and losing points is like using negative ones. If you gain 3 points and then lose 2, your total change is similar to saying you have lost a net of 5 points.

(ii) Take (–3)– 4 × (–3)–3 = (−3)4 + (−3) = (−3)–7.

This example demonstrates the application of the exponent rule to a negative base. Here, if you multiply $(-3)^{-4}$ and $(-3)^{-3}$, you add the exponents. Thus, it simplifies to $(-3)^{-7}$. This confirms our initial rule of exponent addition still applies, regardless of the base being positive or negative.

Think of a thermometer: moving above zero temperatures is like positive exponents, while moving below zero is like negative exponents. Combining both can give you a clearer picture of temperature changes, similar to adding exponents.

In general, we can say that for any non-zero integer a, am × an = am + n, where m and n are integers.

This statement summarizes the law of exponents for all integers, confirming it works for both natural and negative integers. The formula emphasizes that when multiplying powers with the same base, the exponents can be added, which is crucial for simplification in mathematics.

Think of adding weights: if you have weights of 3 kg and 2 kg, when combined they give you a total of 5 kg. Similarly, with exponents, combining them gives a total exponent.

On the same lines, you can verify the following laws of exponents, where a and b are non-zero integers and m, n are any integers.

This chunk mentions additional laws of exponents which can be verified using previous principles. For example, the laws state that dividing powers with the same base involves subtracting the exponents, and raising a power to another power means multiplying the exponents.

Consider laws of exponents as traffic rules: they help streamline calculations, simplifying complex processes into straightforward ones, just like following traffic rules makes travel safer and smoother.

Let us solve some examples using the above Laws of Exponents.

In this part, we can see specific problems and solutions that illustrate the laws of exponents in action. Each example not only applies the laws but also demonstrates how they simplify expressions.

Think of solving problems in math as navigating through a maze. Each law is like a map that helps you find the quickest route to the solution, making complex tasks manageable.

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