Year 10 Extended · Unit 2
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Index laws
10.2.1 · 1 week
Can we generalise each rule? Can we prove it?
Why? Show each with a numerical example.
| \(a^m \times a^n = a^{m+n}\) | \(2^3 \times 2^4 = 2^{3+4} = 2^7\) |
| \(a^m \div a^n = a^{m-n}\) | \(3^5 \div 3^2 = 3^{5-2} = 3^3\) |
| \((a^m)^n = a^{mn}\) | \((2^3)^4 = 2^{3\times 4} = 2^{12}\) |
| \((ab)^n = a^nb^n\) | \((2 \times 3)^3 = 2^3 \times 3^3\) |
| \(a^0 = 1\) | \(5^3 \div 5^3 = 5^0 = 1\) |
| \(a^{-n} = \frac{1}{a^n}\) | \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\) |
| \(a^{1/n} = \sqrt[n]{a}\) | \(9^{1/2} = \sqrt{9} = 3,\quad 8^{1/3} = \sqrt[3]{8} = 2\) |
| \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\) | \(8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4\) |
Tinted amber: new this year.
Simplify \((3a^2b)^3\)
Simplify \(20p^6q^4 \div 4p^2q\)
Each line is wrong. Find the mistake and correct it.
Mini-whiteboards. Answer, then hold up.
Evaluate \(\left(\frac{2}{3}\right)^{-2}\)
Compare \(6x^{-2}\) and \((6x)^{-2}\)
Evaluate \(\left(\frac{3}{4}\right)^{-2}\)
Mini-whiteboards. Answer, then hold up.
| Power | Value | As a fraction |
|---|---|---|
| \(2^4\) | 16 | |
| \(2^3\) | 8 | |
| \(2^2\) | 4 | |
| \(2^1\) | ||
| \(2^0\) | ||
| \(2^{-1}\) | ||
| \(2^{-2}\) | ||
| \(2^{-3}\) | ||
| \(2^{-4}\) |
Label the base and the exponent in \(m^{-a}\). What does the negative sign do?
Raising a number to the power \(\frac{1}{2}\) is the same as finding its square root.
We know that: \(\sqrt{a} \times \sqrt{a} = a\).
We also know that: \(a^{1/2} \times a^{1/2} = a^{\frac{1}{2}+\frac{1}{2}} = a^1 = a\).
Since \(\sqrt{a} \times \sqrt{a} = a\) and \(a^{1/2} \times a^{1/2} = a\), \(\Rightarrow \sqrt{a} \equiv a^{1/2}\).
Prove that \(a^{1/3} = \sqrt[3]{a}\) in the same way.
\(a^{m/n} = \left(\sqrt[n]{a}\right)^m\)
\(a^{m/n} = \left(\sqrt[n]{a}\right)^m\)
Evaluate \(27^{2/3}\)
Evaluate \(16^{-3/4}\)
Evaluate \(125^{-2/3}\)
Write \(20^{2/3}\) as a root.
Mini-whiteboards. Answer, then hold up.
Mini-whiteboards. Answer, then hold up.
Simplify \((64a^6)^{-2/3}\), giving your answer with a positive index.
Simplify \((27x^6)^{-2/3}\), giving your answer with a positive index.
Simplify \(\frac{10x^{-2}}{2x^{-5}}\), giving your answer with a positive index.
Simplify \((27x^6y^{-3})^{2/3}\), giving your answer with positive indices.
Write \(\frac{5}{\sqrt[3]{x^2}}\) in the form \(ax^n\).
Simplify \(\left(\frac{8a^{-3}}{27b^6}\right)^{-2/3}\), giving your answer with positive indices.
Write \(\frac{x^2 + 3\sqrt{x}}{x^{3/2}}\) as a sum of terms of the form \(ax^n\).
Write \(\frac{2x^3 - 5\sqrt{x}}{4x^2}\) in the form \(ax^p + bx^q\).
Simplify fully \(\frac{\sqrt{25x^3y^5}}{10xy^6}\), giving your answer using indices.
Expand and simplify \(\left(3 + \frac{\sqrt{x}}{2}\right)^2\), writing every term in the form \(ax^k\).
Expand and simplify \((x^{1/2} + x^{-1/2})(x^{1/2} - x^{-1/2})\).
Given that \(a^n = \frac{\sqrt[4]{a^3}}{\sqrt[3]{a} \times \sqrt[3]{a^2}}\), find the value of \(n\).
Show that \(\frac{3^{n+2} - 3^n}{8 \times 3^{n-1}} = 3\) for every value of \(n\).
Mini-whiteboards. Answer, then hold up.
Simplify fully \(\left(\frac{9t^4w^9}{18t^6w^{10}}\right)^{-2}\).
Edexcel IGCSE
Write \(\left(\frac{m^6k^{10}}{25}\right)^{3/2}\) in the form \(\frac{m^ak^b}{c}\).
Simplify fully \(\frac{x^{3/2}\,(x^2 - 4)}{x^{1/2}\,(x + 2)}\).
Expand and simplify \(\left(x^{1/2} - 2x^{-1/2}\right)^2\).
\(\left(\frac{27x^9}{y^6}\right)^{-2/3}\) can be written as \(\frac{x^a y^b}{c}\), where \(a\), \(b\), \(c\) are integers. Determine \(a\), \(b\) and \(c\).
Show that \(\frac{2^{n+3} \times 4^n}{8^{n+1}}\) has the same value for every \(n\), and state that value.