What is a^{\frac{m}{n}} in root form?

\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m.

Is \sqrt{x+y}=\sqrt x+\sqrt y?

No. Roots split over products, not sums.

What does 'rationalise the denominator' mean?

Remove surds from the denominator.

State the difference of two squares for surds.

(\sqrt x+\sqrt y)(\sqrt x-\sqrt y)=x-y.

Laws of indices and surdsEdexcel A-Level Maths: Flashcards

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State the three laws of indices.

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State the three laws of indices.: $a^m\times a^n=a^{m+n}$ , $\frac{a^m}{a^n}=a^{m-n}$ , $(a^m)^n=a^{mn}$ .
What is $a^0$ ?: 1
What does $a^{-n}$ mean?: $\frac{1}{a^n}$ .
What is $a^{\frac{m}{n}}$ in root form?: $\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m$ .
Evaluate $16^{-\frac{3}{4}}$ .: $\frac{1}{8}$ .
Evaluate $\left(\frac{8}{27}\right)^{-\frac{2}{3}}$ .: $\frac94$ .
Simplify $\sqrt{18}$ .: $3\sqrt2$ .
What is $(\sqrt x)^2$ ?: $x$ (for $x\geqslant0$ ).
Is $\sqrt{x+y}=\sqrt x+\sqrt y$ ?: No. Roots split over products, not sums.
What does 'rationalise the denominator' mean?: Remove surds from the denominator.
What do you multiply by to rationalise $\frac{1}{3+\sqrt5}$ ?: $\frac{3-\sqrt5}{3-\sqrt5}$ .
State the difference of two squares for surds.: $(\sqrt x+\sqrt y)(\sqrt x-\sqrt y)=x-y$ .
Express $2^{x+3}$ in terms of $a=2^x$ .: $8a$ .
Rationalise $\frac{6}{\sqrt3}$ .: $2\sqrt3$ .

Exam questions on Laws of indices and surds

The number $N$ is given by $N=\left(\frac{8}{27}\right)^{-\frac{2}{3}}$ .
Find the exact value of $N^{-\frac{1}{2}}$ .2 marks
It is given that $2^{x}=a$ , where $a>0$ .
Express $2^{3x-1}$ in terms of $a$ .2 marks
A rectangle has width $(3+\sqrt{5})$ cm and area $(11+5\sqrt{5})$ cm $^2$ .
Show that the length of the rectangle is $(2+\sqrt{5})$ cm.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).