Simplifying Indices Made Easy with Index Law of Subtraction

As an experienced mathematics tutor, I have seen many students struggle with the concept of indices and the various laws that govern them. One of the most important laws in this area is the Index Law of Subtraction, which can help simplify complex expressions and make problem-solving much easier. In this article, we will explore the Index Law of Subtraction in detail, understand its applications, and learn how to use it effectively.

What is the Index Law of Subtraction?

The Index Law of Subtraction is a fundamental rule that allows us to simplify expressions involving indices by subtracting their powers. Formally, it states that for any non-zero real numbers $a$ and $b$, and any real numbers $m$ and $n$:

$\displaystyle \frac{a^m}{a^n} = a^{m-n}$

In other words, when dividing two expressions with the same base, we can subtract the index of the denominator from the index of the numerator.

Example 1

Let’s consider a simple example to understand how the Index Law of Subtraction works. Simplify the following expression:

$\displaystyle \frac{3^5}{3^2}$

Using the Index Law of Subtraction, we can rewrite this as:

$3^{5-2} = 3^3 = 27$

Combining the Index Law of Subtraction with Other Index Laws

The Index Law of Subtraction can be used in conjunction with other index laws to simplify more complex expressions. Let’s take a look at some of these combinations.

Index Law of Multiplication

The Index Law of Multiplication states that when multiplying two expressions with the same base, we can add their indices:

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

Example 2

Simplify the following expression using the Index Laws of Multiplication and Subtraction:

$\displaystyle \frac{2^3 \times 2^4}{2^5}$

Step 1: Use the Index Law of Multiplication to simplify the numerator.
$2^3 \times 2^4 = 2^{3+4} = 2^7$

Step 2: Apply the Index Law of Subtraction to simplify the expression further.
$\displaystyle \frac{2^7}{2^5} = 2^{7-5} = 2^2 = 4$

Index Law of Zero

The Index Law of Zero states that any non-zero number raised to the power of zero is equal to 1:

$a^0 = 1, \text{ where } a \neq 0$

Example 3

Simplify the following expression using the Index Laws of Zero and Subtraction:

$\displaystyle \frac{5^3}{5^3}$

Using the Index Law of Subtraction, we can rewrite this as:

$5^{3-3} = 5^0 = 1$

Negative Indices

Negative indices can be treated using the Index Law of Subtraction as well. A negative index indicates the reciprocal of the base raised to the positive value of the index:

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

Example 4

Simplify the following expression using the Index Laws of Subtraction and Negative Indices:

$\displaystyle \frac{4^{-2}}{4^{-5}}$

Step 1: Rewrite the expression using the Negative Indices rule.
$\displaystyle \frac{\frac{1}{4^2}}{\frac{1}{4^5}}$

Step 2: Simplify the fraction by dividing the numerator by the denominator.
$\displaystyle \frac{1}{4^2} \times \frac{4^5}{1} = \frac{4^5}{4^2}$

Step 3: Apply the Index Law of Subtraction to simplify the expression further.
$\displaystyle \frac{4^5}{4^2} = 4^{5-2} = 4^3 = 64$

Simplifying Complex Expressions

Now that we have explored the Index Law of Subtraction and its combinations with other index laws, let’s apply this knowledge to simplify a more complex expression.

Example 5

Simplify the following expression:

$\displaystyle \frac{6^2 \times 6^{-3}}{6^4 \times (6^{-1})^3}$

Step 1: Simplify the numerator using the Index Law of Multiplication.
$6^2 \times 6^{-3} = 6^{2+(-3)} = 6^{-1}$

Step 2: Simplify the denominator using the Index Laws of Multiplication and Negative Indices.
$6^4 \times (6^{-1})^3 = 6^4 \times 6^{-3} = 6^{4+(-3)} = 6^1 = 6$

Step 3: Apply the Index Law of Subtraction to simplify the expression further.
$\displaystyle \frac{6^{-1}}{6} = 6^{-1-1} = 6^{-2} = \frac{1}{6^2} = \frac{1}{36}$

Conclusion

The Index Law of Subtraction is a powerful tool for simplifying expressions involving indices. By understanding its applications and how it can be combined with other index laws, you can tackle even the most complex problems with ease. Remember to always look for opportunities to apply the Index Law of Subtraction when dividing expressions with the same base, and practice regularly to solidify your understanding of this essential concept.

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