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Multiplying Indices

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Learning text on the topic Multiplying Indices

Multiplying Indices

Indices are more than just numbers on paper; they're a key part of the maths you'll encounter in and out of school. Think of indices as a shortcut for repeated multiplication, simplifying how we handle larger numbers. In everyday life, you see them in situations like calculating how fast a popular video goes viral or understanding how interest on your savings account works. Let's explore multiplying powers with the same base! We'll uncover the logic behind the 'keep the base, add the indices' rule through easy expansions, making this concept simple to grasp.

Multiplying Indices – Rules

To grasp why we add indices, or exponents, when multiplying powers of the same base, it's helpful to look at the process of expansion.

When multiplying powers with the same base, keep the base the same and find the sum of the exponents, or indices.

Let's use concrete numbers for clarity. Consider $2^3 \cdot 2^2$. Here, $2$ is the base, and $3$ and $2$ are the exponents. We can expand these terms to understand the addition of exponents:

  • $2^3$ means multiplying $2$ by itself $3$ times, which is $2 \cdot 2 \cdot 2$.
  • $2^2$ means multiplying $2$ by itself $2$ times, which is $2 \cdot 2$.

When you multiply $2^3$ and $2^2$, you're essentially multiplying $2$ by itself $3 + 2=5$ times. Therefore, $2^3 \cdot 2^2 = 2^{(3+2)} = 2^5$.

27601_Multiplication_with_Exponents-01.svg

Multiplying Indices – Different Base

When dealing with expressions that have different bases and indices, such as $a^x$ and $b^y$, it's essential to treat each base and its index separately. Here, $a$ and $b$ represent different bases, while $x$ and $y$ are their respective indices.

For different bases, calculate each exponential term individually. The multiplication of $a^x$ and $b^y$ can be expressed as $a^x \times b^y = (a^x) \times (b^y)$. Each base is raised to its index, and the results are then multiplied.

In practice, this means you should first evaluate $a^x$ and $b^y$ separately and then multiply these outcomes to find the final result.

27601_-_Multiplication_with_Exponents-04.svg

Multiplying Indices – Different Base, Same Exponent

When multiplying indices that have different bases but the same exponent, the process is slightly different from other scenarios in index multiplication. This situation often arises in various mathematical contexts and requires a distinct approach.

When multiplying powers with different bases but the same index, you multiply the bases first and then apply the common index to the result.

To understand this, consider an example like $2^3 \cdot 3^3$. Here, $2$ and $3$ are different bases, but both have the same index, $3$. The multiplication can be approached by first multiplying the bases and then applying the index:

  • First, multiply the bases: $2 \times 3 = 6$.
  • Then, apply the common index: $6^3$.

So, $2^3 \cdot 3^3 = 6^3$.

Below we can see another example.

27601_-_Multiplication_with_Exponents-03.svg

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.

See how much you understand about multiplying exponents so far with this interactive quiz!

Expand and simplify $4^2 \cdot 4^5$.
Expand and simplify $8^5 \cdot 8^7$.

Multiplying Indices – Example

Let’s see how the above explanation looks in practice. In the following table, you will see step by step progression from a multiplication equation, to expansion, to the addition of the indices and finally simplification:

Multiply Expansion Add the Indices Simplify
$6^3 \cdot 6^5$ $6 \cdot 6 \cdot 6) \cdot (6 \cdot 6 \cdot 6 \cdot 6 \cdot 6)$ $6^{(3+5)}$ $6^8$
$n^2 \cdot n^4$ $(n \cdot n) \cdot (n \cdot n \cdot n \cdot n)$ $n^{(2+4)}$ $n^6$

The process of multiplying indices can be used in some real-world problems, such as multiplying and dividing numbers in scientific notation, or standard form, which is helpful when working with very large or very small numbers.

Multiplying Indices – Application

Multiply $4^3 \cdot 4^2$ and explain each step.
Expand and simplify $x^2 \cdot x^4$.
What value for $x$ would make this true? $4^3 \cdot 4^x = 4^{10}$.
Identify the misconception: A student calculated $5^3 \cdot 5^3 = 25^9$. What misconception did they have?

Tips for Multiplying Indices

  • Understand the Base: Remember that the base number remains the same when multiplying indices.
  • Expand for Clarity: Breaking down the expression by expanding each term can help visualise why the indices are added.
  • Add the Exponents: Focus on adding the indices when the base is the same. This is key to simplifying the expression correctly.
  • Check Your Work: After simplifying, it's always good practice to check your work by expanding the expression to ensure it matches the original values.

Multiplying Indices – Extension

In multiplying indices, another important concept is the Power of a Power law. This rule comes into play when you have an index raised to another index, like $(n^a)^b$.

Power of a Power Law

The Power of a Power law states that when you raise a power to another power, you multiply the indices. This is written as $(n^a)^b = n^{a \cdot b}$.

For example, $(3^2)^4$can be expanded as $3^{2 \cdot 4} = 3^8$. This shows that you multiply the index outside the brackets with the index inside.

27601_Multiplication_with_Exponents-02.svg

Understanding the Power of a Power law is crucial as it simplifies complex expressions and is a key component in algebra and higher-level maths.

Multiplying Indices – Summary

Let’s review what we learned about multiplication with indices in this text.

Key Points from This Text:

  • To find the product of indices with the same base, keep the base and add the indices.
  • This rule is based on expanding the exponential terms and counting the total number of times the base is multiplied.
  • Mastery of this index multiplication is essential for advancing in algebra and other mathematical areas.

Feel free to complete interactive practice problems on this topic or complete our worksheets.

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

Multiplying Indices – Frequently Asked Questions

What is an index?
Why do we add indices when multiplying terms with the same base?
Can this rule be applied to bases that are different?
What happens if the indices are negative?
Is this rule applicable to real-world problems?
What if one of the indices is zero?
How does this rule help in simplifying algebraic expressions?
Can this index rule be used in equations?
What is the importance of understanding index rules in higher mathematics?
How can students practise this concept effectively?
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