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Stage 5.1

1.01 The index laws

Lesson

Index laws

Here is a summary of the index laws we know so far:

The multiplication law: a^m\times a^n=a^{m+n}

The division law: a^m\div a^n=\dfrac{a^m}{a^n}=a^{m-n}, a\neq 0

The power of a power law: \left(a^m\right)^n=a^{m\times n}

The zero power law: a^0=1,a\neq 0

Combining one or more of these laws, together with the order of operations, we can simplify more complicated expressions involving powers. Let's look at some examples to see how.

Examples

Example 1

Fill in the blank to make the equation true.11^{11} \times 11^{⬚}=11^{19}

Worked Solution
Create a strategy

Use the multiplication law: a^m\times a^n=a^{m+n}.

Apply the idea

Since we are given m+n=19 and m=11, then we find n.

\displaystyle 11+n\displaystyle =\displaystyle 19Use the multiplication law
\displaystyle 11+n-11\displaystyle =\displaystyle 19-11Subtract 11 on both sides
\displaystyle n\displaystyle =\displaystyle 8Evaluate

Example 2

Simplify \dfrac{\left(17^5\right)^8}{17^{32}}, giving your answer in index form.

Worked Solution
Create a strategy

Use the power of a power law and then the division law to simplify the expression.

Apply the idea
\displaystyle \frac{\left(17^5\right)^8}{17^{32}}\displaystyle =\displaystyle \frac{17^{5\times 8}}{17^{32}}Use the power of a power law
\displaystyle =\displaystyle \frac{17^{40}}{17^{32}}Evaluate the product
\displaystyle =\displaystyle 17^{40-32}Use the division law
\displaystyle =\displaystyle 17^8Evaluate the subtraction
Idea summary

Summary of index laws:

The multiplication law: a^m\times a^n=a^{m+n}

The division law: a^m\div a^n=\dfrac{a^m}{a^n}=a^{m-n}, a\neq 0

The power of a power law: \left(a^m\right)^n=a^{m\times n}

The zero power law: a^0=1,a\neq 0

Negative bases

The same rules apply when we are dealing with negative bases, we just need to take care if we are asked to evaluate. We know that the product of two negative numbers is positive, and the product of a positive and a negative number is negative. This means we need to be extra careful when evaluating powers of negative bases.

A negative base raised to an even power will evaluate to a positive answer.

  • For example \left(-3\right)^4=3^4=81

    • \left(-3\right)^4\neq -3^4

A negative base raised to a odd power will evaluate to a negative answer.

  • For example \left(-2\right)^5=-2^5=32

    • \left(-2\right)^5=-2^5

Examples

Example 3

Using index laws, evaluate \left(-4\right)^{11}\div \left(-4\right)^7.

Worked Solution
Create a strategy

Use the division law.

Apply the idea
\displaystyle \left(-4\right)^{11}\div \left(-4\right)^7\displaystyle =\displaystyle \left(-4\right)^{11-7}Use division law
\displaystyle =\displaystyle \left(-4\right)^4Evaluate the subtraction
\displaystyle =\displaystyle 256Evaluate
Idea summary

A negative base raised to an even power will evaluate to a positive answer. (-3)^2=9

A negative base raised to a odd power will evaluate to a negative answer. (-3)^3=-27

Fractional bases

When raising a fractional base, we apply the power to both the numerator and the denominator.

Let's consider a simple example like \left(\dfrac{1}{2}\right)^2. This expands to \dfrac{1}{2}\times \dfrac{1}{2}=\dfrac{1\times 1}{2\times 2}, which evaluates to \dfrac{1}{4} or \dfrac{1^2}{2^2}.

Similarly, a slightly harder expression like \left(\dfrac{2}{3}\right)^3expands to \dfrac{2}{3}\times \dfrac{2}{3}\times \dfrac{2}{3} giving us \dfrac{2\times 2\times 2}{3\times 3\times 3}. So we can see that \left(\dfrac{2}{3}\right)^3=\dfrac{2^3}{3^3}.

This can be generalised to give us the following rule:

For any base number of the form \dfrac{a}{b}, and any number n as a power, \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}

Examples

Example 4

Simplify \left(\dfrac{23}{41}\right)^8.

Worked Solution
Create a strategy

Use the property \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}.

Apply the idea
\displaystyle \left(\dfrac{23}{41}\right)^8\displaystyle =\displaystyle \dfrac{23 ^{8}}{41^{8}}Apply the power property
Idea summary

For any base number of the form \dfrac{a}{b}, and any number n as a power, \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}

Outcomes

MA5.1-5NA

operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases

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