Division with Indices

• Expand the term: $4^{6} = 4 × 4 × 4 × 4 × 4 × 4$
• Expand the term: $4^2 = 4 × 4$.
• In division, cancel the two $4$s in $4^2$ from the six $4$s in $4^6$.
• You are left with four $4$s, or $4^4$.
• Therefore, $4^6 ÷ 4^2 = 4^{(6-2)} = 4^4$.

• Expand the terms: $5^7 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$, and $5^4 = 5 \cdot 5 \cdot 5 \cdot 5$
• Cancel the four $5$s in $5^4$ from the seven $5$s in $5^7$
• This leaves us with three $5$s, or $5^3$
• So, $\dfrac{5^7}{5^4} = 5^{(7-4)} = 5^3$

When dividing indices with the same base, we subtract the indices. Here, $8^5 \div 8^x = 8^{(5-x)}$. Since this equals $8^2$, the indices must be the same. So, $5 - x = 2$. Using our knowledge of integers, $x$ must be $3$ because subtracting $3$ from $5$ gives $2$.

The misconception lies in the understanding of base and index treatment during division. The correct rule for dividing indices with the same base is to subtract the indices while keeping the base unchanged. The accurate calculation should be $7^4 \div 7^2 = 7^{(6-2)} = 7^4$, not $49^3$. They incorrectly squared the base from $7$ to $49$ while also misapplying the index operation. Another misconception that could occur is dividing $7$ by $7$ to get $1$ and then dividing $6$ by $2$ to get $3$ leaving an answer of $1^3$ which would be incorrect. The accurate calculation should be $7^4 \div 7^2 = 7^{(6-2)} = 7^4$.

Dividing indices refers to the process of dividing two exponential expressions that have the same base. The rule is to keep the base and subtract the indices.

No, the rule for dividing indices only applies when the bases are the same. For different bases, other methods or rules need to be used.

To simplify $5^7 \div 5^4$, subtract the index of the divisor $(4)$ from the index of the dividend $(7)$. This gives $5^{(7-4)} = 5^3$.

If the indices are the same, such as in $2^5 \div 2^5$, the expression simplifies to $2^0$, which equals $1$.

Yes, if the index of the divisor is larger than the index of the dividend, the result will be a negative index. For example, $3^2 \div 3^5 = 3^{(2-5)} = 3^{-3}$.

Yes, dividing indices is useful in real-world scenarios, such as in scientific calculations involving rates of decay or growth, or in financial calculations involving compound interest.

A common mistake is to divide the bases instead of keeping the base the same and subtracting the indices.

Practice dividing indices by solving various problems involving exponential expressions with the same base. Use both numerical and algebraic expressions for a better understanding.

Learning about dividing indices is crucial as it forms a foundational skill in algebra, necessary for higher-level maths and various scientific fields.

Yes, dividing indices is a key concept in understanding exponential growth and decay, which are common in fields like biology, chemistry and physics.