# Division with Indices

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## Dividing Indices

In maths, indices, or exponents, are more than just numbers; they're keys to unlocking puzzles in the world around us. From calculating the decay of radioactive materials in science class to understanding how quickly a population of endangered species might decrease, Dividing Indices is an essential tool. This guide focuses on how to effectively divide numbers with the same base and their indices. We'll delve into this concept with relatable examples, demonstrating the rule: keep the base, or power, the same and subtract the indices.

If you are finding it hard to follow the text, we recommend consolidating your understanding of powers by watching this video: Understanding Powers of 10.

## Understanding How to Divide Indices

The concept of dividing numbers written in standard form that have the same base follows a straightforward rule.

When dividing numbers written in standard form, or exponential notation, that have the same base, keep the base the same and subtract the indices.

For a clear example, let's look at this problem: $3^5 ÷ 3^2$.

Here, $3$ is the base, and $5$ and $2$ are the indices. By expanding these terms, we can see the logic behind subtracting the indices:

• $3^5 = 3 × 3 × 3 × 3 × 3$
• $3^2 = 3 × 3$

When dividing $3^5 ÷ 3^2$, we eliminate the two $3$s in $3^2$ from the five $3$s in $3^5$. This leaves us with three $3$s, or $3^3$. Thus, the expression simplifies to $3^{(5-2)} = 3^3$.

## Dividing Indices – Steps to Simplify

Divide Expansion Subtract the Indices Simplify
$6^5 \div 6^3$ $(6 × 6 × 6 × 6 × 6) \div (6 × 6 × 6)$ $6^{(5-3)}$ $6^2$
$\dfrac{(n^5)}{(n^4)}$ $\dfrac{(n × n × n × n × n)}{(n × n × n × n)}$ $n^{(5-4)}$ $n^1 = n$

Let's simplify some examples:

Expand first, and then simplify. $4^{6} ÷ 4^{2}$
Expand first, and then simplify. $\dfrac{5^7}{5^4}$

## Dividing Indices – Application

What value for $x$ would make this true? $8^5 \div 8^x = 8^2$.
Identify the misconception: A student calculated $7^6 \div 7^2 = 49^3$. What misconception did they have?

## Dividing Indices – Summary

Key Points from This Text:

• Dividing indices with the same base involves subtracting the index of the divisor from the exponent of the dividend.
• This rule simplifies the process of working with exponential expressions in division.
• Understanding this concept is crucial for both academic purposes and real-world applications, like analysing trends or growth rates.

Explore other topics, interactive problems, videos and printable worksheets on our website to further enhance your understanding of Division with Indices and other vital mathematical concepts.

To explore exponents further, have a look at Exponents with Negative Bases and Multiplication with Indices.

## Dividing Indices – Frequently Asked Questions

What does it mean to divide indices?
Can you divide indices with different bases?
How do you simplify an expression like $5^7 \div 5^4$?
What if the indices in the division are the same?
Is it possible to have a negative index after dividing indices?
Can this rule be used in real-world problems?
What is a common mistake when dividing indices?
How can I practise dividing indices?
Why is it important to learn about dividing indices?
Can dividing indices be applied in understanding exponential growth and decay?

## Division with Indices exercise

Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the learning text Division with Indices.
• ### Explain the rule for dividing indices.

Hints

Notice in this example, the matching base is the variable $n$.

This stays the same.

Another way to understand dividing indices is to expand each index. Then it will make it clear that subtraction is used to find the final index.

Solution

When dividing indices that have the same base, the rule is keep the base the same and subtract the indices.

For example, $\dfrac{3^7}{3^3}$, the base is $\bf{3}$, so this stays the same. To find the quotient's index, subtract the indices: $\bf{7-3}$$=4$. The quotient is equal to $\bf{3^4}$.

• ### How else could we show this expression?

Hints

To expand an index, the base is multiplied by itself the number of times the value of the index is.

For example, $4^3$ = $4 \cdot 4 \cdot 4$.

$\dfrac{5^6}{5^4}$

For the numerator, the base is $5$, and there are $6$ of them being multiplied. So, $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$.

What is the base of the denominator, and how many times is that number being multiplied by itself?

$\cdot$ is used as a symbol for multiplication.

Solution

$\dfrac{5^6}{5^4}$

The numerator has a base of $5$, and since the index is $6$, the $5$ is expanded as: $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$.

The denominator has a base of $5$, and the index is $4$, so the $5$ is expanded as: $5\cdot 5\cdot 5\cdot 5$

$\dfrac{5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}{5\cdot 5\cdot 5\cdot 5}$

• ### Divide the indices using the rule.

Hints

If you divide two numbers with the same base, you just subtract the smaller index from the bigger one and keep the base the same. For example, $a^m \div a^n = a^{m-n}$ when $a$ is a number and $m$ and $n$ are the indices.

$\dfrac{2^7}{2^6}$

$2^{7-6}$

$2^1=2$

Solution

$\dfrac{5^9}{5^3}$

$5^{9-3}$

$5^6$

• ### Apply the rule to evaluate expressions.

Hints

When you divide two numbers with the same base, you keep the base the same and subtract the indices.

If it helps you to better understand the rule, first expand the expression and then cross off pairs.

Solution
• $\frac{4^7}{4^3} = 4^{7-3} = 4^4$
• $\frac{4^4}{4} = 4^{4-1} = 4^3$
• $\frac{3^6}{3^2} = 3^{6-2} = 3^4$
• $\frac{4^8}{4^2} = 4^{8-2} = 4^6$
• ### Apply the rule for dividing indices.

Hints

When you divide two numbers with the same base, you keep the base the same and subtract the indices.

The base in this example is $3$, and then the indices are subtracted, $6-5$.

Solution

The base is $2$.

The indices will be subtracted $8-2$.

The final solution will be $2^6$.

• ### Find the solution by applying the rule.

Hints

When you type in your answer, use the $x^a$ button on the toolbar to create an index.

If you divide two numbers with the same base, you just subtract the smaller index from the bigger one and keep the base the same.

For example, $a^m \div a^n = a^{m-n}$ when $a$ is a number and $m$ and $n$ are the indices.

Solution

$\begin{array}{l}\frac{5^5}{5^2}=5^{5-2}=5^3\\ \\ \frac{8^2}{8}=8^{2-1}=8\\ \\ \frac{9^7}{9^2}=9^{7-2}=9^5\\ \\ \frac{x^6}{x^4}=x^{6-4}=x^2\end{array}$

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