Quick questions on Indices and surds: laws of indices, rationalising denominators - OCR A-Level Maths A

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are surds?

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A surd is an irrational root such as

\sqrt{2}

\sqrt[3]{5}

. The key manipulation rules are

\sqrt{ab} = \sqrt{a}\,\sqrt{b}

and

\sqrt{\tfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}

. To simplify a surd, take out the largest square factor:

\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}

What is rationalising the denominator?

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A fraction is "rationalised" when no surd appears in the denominator. For a single surd, multiply top and bottom by that surd. For a denominator

a + b\sqrt{c}

, multiply by the conjugate

a - b\sqrt{c}

, because

(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2 c

is rational.

What is solving an equation with a hidden quadratic in a power?

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Index laws let you spot a "hidden quadratic" when an unknown appears in an exponent, for example

4^x - 5(2^x) + 4 = 0

. Writing

y = 2^x

turns

4^x = (2^x)^2 = y^2

, so the equation becomes the quadratic

y^2 - 5y + 4 = 0

. This substitution trick recurs in the exponentials topic too.

What is q1?

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Simplify

\sqrt{50} + \sqrt{18}

. [2 marks]

What is q2?

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Rationalise and simplify

\dfrac{6}{\sqrt{3}}

. [2 marks]

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