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Algebraic Roots & Indices (Cambridge (CIE) IGCSE Maths)

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Write down the index law for $a^{m} \times a^{n}$ .

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Write down the index law for $a^{m} \times a^{n}$ .
The index law is: $a^{m} \times a^{n} = a^{m + n}$ .
You add the powers when multiplying two powers with the same base.
E.g. $a^{3} \times a^{5} = a^{8}$ .
Write down the index law for $\frac{a^{m}}{a^{n}}$ .
The index law is: $\frac{a^{m}}{a^{n}} = a^{m - n}$ .
You subtract the powers when dividing two powers with the same base.
E.g. $\frac{f^{9}}{f^{3}} = f^{6}$ .
Write down the index law for ${(a^{m})}^{n}$ .
The index law is: ${(a^{m})}^{n} = a^{m n}$ .
You multiply the powers when raising $a^{m}$ to the power $n$ .
E.g. ${(g^{4})}^{8} = g^{32}$ .
Write down the index law for $a^{\frac{1}{m}}$ .
The index law is: $a^{\frac{1}{m}} = \sqrt[m]{a}$ .
A fractional power of $\frac{1}{m}$ means the $m^{th}$ root.
E.g. $x^{\frac{1}{5}} = \sqrt[5]{x}$ .
Write down the index law for ${(a b)}^{n}$ .
The index law is: ${(a b)}^{n} = a^{n} b^{n}$ .
The power outside acts on every term inside, when the terms inside are multiplied together.
E.g. ${(t y)}^{3} = t^{3} y^{3}$ .
Write down the index law for $a^{- m}$ .
The index law is: $a^{- m} = \frac{1}{a^{m}}$ .
A negative power means 1 over the positive power.
E.g. $h^{- 2} = \frac{1}{h^{2}}$ .
Write down the index law for $a^{1}$ .
The index law is: $a^{1} = a$ .
Anything to the power 1 is itself.
E.g. $w^{1} = w$ .
Write down the index law for $a^{0}$ .
The index law is: $a^{0} = 1$ .
Anything to the power 0 is 1.
E.g. $y^{0} = 1$ .
True or False?
$a^{\frac{m}{n}} = \sqrt[m]{a^{n}}$
False.
The order of $m$ and $n$ are incorrect.
The correct index law is $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ .
E.g. $x^{\frac{3}{4}} = \sqrt[4]{x^{3}}$ .
True or False?
$a^{\frac{m}{n}}$ can be thought of in two different ways, using powers and roots.
True.
You can swap the order of the $n^{th}$ root and the power of $m$ .
So $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ is the first way to think about it, and $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ is the second way to think about it.
E.g. $y^{\frac{3}{5}} = \sqrt[5]{y^{3}} = {(\sqrt[5]{y})}^{3}$ .
When talking about indices, what is the base?
The base is the number or letter being acted on by the power.
For example, in $a^{n}$ , the base is $a$ and the power is $n$ .
What is the first step to solving the equation $3^{x} = 27^{x + 1}$ ?
You need both sides to be over the same base.
So the first step is to change the base of 27 to 3, using the fact 27 = 3³.
(Alternatively, you could change the base of 3 to 27, as $3 = 27^{\frac{1}{3}}$ ).
True or False?
${(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n}$
True.
A fraction to a negative power is the same as the flipped fraction to the positive power, ${(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n}$ .
This is a great trick to use in exams!

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