01.01 Laws of Indices

VIDEO LESSON

\(\displaystyle \require{cancel} \frac{a \cancel{b}}{\cancel{b}} = a\)

INTERACTIVE SELF-STUDY

You should remember laws of indices from GCSE. We will derive each one

• \(a^n\) is called a power. It is the \(n\)th power of \(a\)
• \(a\) is called the base
• \(n\) is the exponent, power or index.

When \(n\) is an integer \(a^n\) can be thought of as repeated multiplication
For example \(a^2=a\times a\), or \(a^4=a\times a \times a \times a\)

In general \(a^n = \underbrace{a \times a \times \cdots \times a \times a}_{n \text{ times}}\)

We will use this idea of repeated multiplication and the number of times \(a\) is multiplied by itself to derive the different laws of indices.

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PRODUCTS AND QUOTIENTS

Consider \(a^m \times a^n\), where \(m\) and \(n\) are integers.
This represents: \((\underbrace{a \times a \times \cdots \times a \times a}_{m \text{ times}})\times (\underbrace{a \times a \times \cdots \times a \times a}_{n \text{ times}})\)

i. How many \(a\)’s are being multiplied together in total? Show Answer

ii. Therefore what is \(a^m \times a^n\) expressed as a single power of \(a\) Show Answer

• \(a^m \times a^n=a^{m+n}\). This is true for any value of \(m\) and \(n\) and not just when they are integers
• This is called the multiplication law or product law

Now consider \(a^m ÷ a^n\), where \(m\) and \(n\) are integers.

This represents: \(\displaystyle \frac{\overbrace{a \times a \times a \times \cdots \times a \times a}^{m \text{ times}}}{\underbrace{a \times a \times \cdots \times a}_{n \text{ times}}}\)

The \(a\)’s in the denominator will cancel out \(n\) of the \(a\)’s in the numerator.
i. How many \(a\)’s will you be left with? Show Answer

ii. Therefore what is \(a^m ÷ a^n\) expressed as a single power of \(a\) Show Answer

• \(a^m ÷ a^n=a^{m-n}\) whenever \(a≠0\). This is true for any value of \(m\) and \(n\) and not just when they are integers
• This is called the division law or quotient law
• A quotient is an expression where one quantity is divided by another.

Consider \(\displaystyle \frac{a^{3}}{a^{2}}\). This represents: \(\displaystyle \frac{a \times a \times a}{a \times a}\)

i. By cancelling, what does this simplify to? Show Answer

ii. Now instead use the division law and express this fraction as a single power of \(a\). Show Answer

• \(a^1=a\). This is called the identity power law

Example 1 – (GRADE E)
Simplify the following expressions
a. \(x^3 \times x^4\) Show Answer

b. \(3p^2 \times 5p^6\) Show Hint Show Answer
c. \(2q^4 \times 3q^7\) Show Answer
d. \(x^8 ÷ x\) Show Answer
e. \(12r^9 ÷ 4r^5\) Show Hint Show Answer
f. \(14m^6 ÷ 2m^3\) Show Answer

Example 2 – (GRADE D)
a. Expand and simplify \(x(4x^2-3x)\) Show Answer

b. Expand and simplify \(4y^3(y^2-5y+7)\) Show Answer
c. Expand and simplify \(5z^2(2z+z^3)-3z(z^2-4)\) Show Hint Show Answer

d. Simplify \(\displaystyle \frac{x^5+x^3}{x^2}\) Show Hint Show Answer

e. Simplify \(\displaystyle \frac{9y^7+3y^4+2y^3}{3y^3}\) Show Answer

f. Simplify \(\displaystyle \frac{12z^6-6z^2}{6z}\) Show Answer

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POWERS OF POWERS, PRODUCTS AND QUOTIENTS

Now consider \((a^m)^n\), where \(m\) and \(n\) are integers.
This represents \(\underbrace{a^m \times a^m \times \cdots \times a^m}_{n \text{ times}}\)

or equivalently \(\underbrace{(\overbrace{a \times a \cdots \times a}^{m \text{ times}}) \times (\overbrace{a \times a \cdots \times a}^{m \text{ times}}) \times \cdots \times (\overbrace{a \times a \cdots \times a}^{m \text{ times}})}_{n \text{ times}}\)

i. How many \(a\)’s are being multiplied together in total? Show Answer

ii. Therefore what is \((a^m)^n\) expressed as a single power of \(a\) Show Answer

• \((a^m)^n =a^{mn}\). This is true for any value of \(m\) and \(n\) and not just when they are integers
• This is called the power of a power law

Consider \((a\times b)^n\), where \(n\) is an integer.
This represents \( \underbrace{(a\times b) \times (a\times b) \times \cdots \times (a\times b) \times (a\times b)}_{n \text{ times}}\)

i. How many \(a\)’s are being multiplied together in total? Show Answer

ii. How many \(b\)’s are being multiplied together in total? Show Answer
ii. Therefore what is \((a\times b)^n\) expressed as a power of \(a\) multiplied by a power of \(b\)? Show Answer

• \((a\times b)^n=a^n \times b^n\). This is called the power of a product law
• It also follows that \(\displaystyle \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}\). This is called the power of a quotient law
• These are true for any value of \(n\) and not just when it is an integer

Example 3 – (GRADE D)
Simplify the following expressions
a. \((x^2)^5\) Show Answer

b. \((xy)^4\) Show Answer
c. \((5x)^3\) Show Answer
d. \(\displaystyle \left(\frac{2}{x}\right)^5\) Show Answer
e. \((4x^5)^3 ÷ (2x^6)^2\) Show Answer

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ZERO AND NEGATIVE POWERS

Consider \(\displaystyle \frac{a^n}{a^n}\)
i. What value should this cancel to? Show Answer

ii. Now instead use the division law and express this fraction as a single power of \(a\). Show Answer

• \(a^0=1\) whenever \(a≠0\). This is called the zero-power law

Consider \(a^{-n} \times a^n\)
i. Use the multiplication law to express this as a single power of \(a\). What is its value? Show Answer

ii. Hence express \(a^{-n}\) in terms of \(a^n\) Show Answer

• \(\displaystyle a^{-n}=\frac{1}{a^{n}}\) whenever \(a≠0\). This is called the negative power law
• It is usually helpful to express negative powers of a variable as fractions as they will be easier to manipulate.

Example 4 – (GRADE D)
Simplify the following expressions
a. \(\displaystyle \frac{x^2}{x^{-4}}\) Show Answer

b. \(\displaystyle \frac{3x^8-2x^6+4x}{x^6}\) Show Answer

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\(n\)-TH ROOTS AND RATIONAL POWERS

Consider the \(n\)th root of \(a\): \(\sqrt[n]{a}\).
We would like to write this as a power of \(a\), that is, express \(\sqrt[n]{a}\) in the form \(a^k\)

We know that if you take \(\sqrt{a}\) and square it, you should get \(a\) back: \((\sqrt{a})^2=a\)
Similarly if you take \(\sqrt[3]{a}\) and cube it, you should get \(a\) back: \((\sqrt[3]{a})^3=a\)

So in general taking the \(n\)th root of \(a\), \(\sqrt[n]{a}\), and raising it to the power of \(n\) should give us back \(a\)
We can write \((\sqrt[n]{a})^n=a\)

i. Suppose \(\sqrt[n]{a}=a^k\). Rewrite \((\sqrt[n]{a})^n=a\) using \(a^k\). Show Answer

ii. Simplify your resulting expression using the power of a power law Show Answer
iii. By comparing the exponents of \(a\) on both sides of the equation, what is \(k\) in terms of \(n\)? Show Answer

• \(\displaystyle \sqrt[n]{a}=a^{\frac{1}{n}}\).
• This is called the \(n\)th root as a power law
• Note that by convention \(\sqrt{a}\) is the positive square root of \(a\). For example \(\sqrt{9}=3\) and not \(−3\)

i. Consider \((\sqrt[n]{a})^m\). Express \(\sqrt[n]{a}\) as a power of \(a\), then use the power of a power law to simplify \((\sqrt[n]{a})^m\) Show Answer

ii. Consider \(\sqrt[n]{a^m}\). Express \(\sqrt[n]{a^m}\) as a power of \(a^m\), then use the power of a power law to simplify \(\sqrt[n]{a^m}\) Show Answer

• \(\displaystyle (\sqrt[n]{a})^m=\sqrt[n]{a^m}=a^{\frac{m}{n}}\).
• This is called the rational power law or power of an \(n\)th root law
• A rational number is a ratio (fraction) of two integers (whole numbers).
  Rational numbers are in the form \(\displaystyle \frac{p}{q}\), where \(p\) and \(q\) are integers.

Example 5 – (GRADE D)
Simplify the following expressions, writing them in the form \(kx^n\) where \(k\) and \(n\) are constants
a. \(\displaystyle \sqrt[4]{x}\) Show Answer

b. \(\displaystyle \sqrt[3]{x^2}\) Show Answer
c. \(\displaystyle (\sqrt[5]{x})^8\) Show Answer
d. \(\displaystyle \left(x^4\right)^{\frac{5}{4}}\) Show Answer
e. \(\sqrt{64x^5}\) Show Hint Show Answer
f. \(6x^{0.5}÷2x^{-2.75}\) Show Answer

Example 6 – (GRADE D)
Evaluate the following using a detailed method
a. \(\displaystyle 49^{\frac{1}{2}}\) Show Answer

b. \(\displaystyle 125^{\frac{1}{3}}\) Show Answer
c. \(\displaystyle 4^{\frac{3}{2}}\) Show Hint Show Answer
d. \(\displaystyle 27^{\frac{2}{3}}\) Show Answer
e. \(\displaystyle 9^{-\frac{5}{2}}\) Show Hint Show Answer

Example 7 – (GRADE C)
Given that \(\displaystyle z=8y^3\), find expressions, in the form \(ky^n\), for:
a. \(\sqrt[3]{z}\) Show Answer

b. \(12z^{-4}\) Show Answer

END OF LESSON!

You can now answer:
EDEXCEL Pure Year 1 Ex 1A – All Questions
EDEXCEL Pure Year 1 Ex 1D – Q1 to Q5

OCR A Student Book 1 Ex 2A – Q5 to Q10, Q14