Laws of Indices & Surds (OCR AS Maths A: Pure): Exam Questions

Show how to write

$\frac{3-2\sqrt{3}}{4-\sqrt{3}}$

in the form

$\frac{a+b\sqrt{3}}{c}$

without using a calculator, where $a$, $b$ and $c$ are integers to be found.

Give your answer in simplest form.

Given that

$y=\frac{1}{16}{x}^4$

write

$y^{\frac{1}{2}}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Given that

$y=\frac{1}{16}{x}^4$

write

$y^{-1}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Given that

$y=\frac{1}{16}{x}^4$

write

$y^{-\frac{3}{2}}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

A student believes that 

$\sqrt{a}+\sqrt{b}=\sqrt{a+b}$

for all values of $a$ and $b$.

By substituting in two different square numbers for $a$ and $b$, show that the student is not correct.

Show how to write 

$\frac{14}{4+\sqrt{2}}$

in the form

$a-\sqrt{b}$

without using a calculator, where $a$ and $b$ are integers to be found.

Solve the equation 

$6-x\sqrt{7}=\frac{2x}{\sqrt{7}}$

giving your answer in the form

$x=\frac{a\sqrt{7}}{b}$

where $a$ and $b$ are non-zero integers to be found and your answer is in simplest form.

You must show the steps in your working clearly.

A rectangle has an area of 14 m² and a length of $\left(3-\sqrt{2}\right)$ m. 

Without using a calculator, find the exact width of the rectangle, showing the steps in your working clearly.

Expand and simplify

$\frac{{\left(3-\sqrt{x}\right)}^2}{x}$

giving each term in simplest form.

Use a non-calculator method to solve the equation

$128\sqrt{2}=2^a$

to find the exact value of $a$.

Show the steps in your working clearly.

Write 

$\frac{x\left(2x^4-\sqrt{x}\right)}{\sqrt{x}}$

in the form

$a{x}^b-x^c$

where $a$, $b$ and $c$ are constants to be found.

Solve the equation

$\frac{1}{9}{x}^{\frac{4}{5}}=9$

showing clearly the steps in your working.

Write

$7x^{-3}÷2x^{-\frac{1}{3}}$

in the form

$a{x}^k$

where $a$ and $k$ are constants to be found.

Write

$\frac{2}{3}{x}^{\frac{7}{6}}\times \frac{5}{2}{x}^{-\frac{5}{3}}$

in the form

$p{x}^q$

where $p$ and $q$ are constants to be found.

Write

${\left(3x^{-\frac{2}{3}}\right)}^2÷6x^{-\frac{1}{6}}$

in the form

$m{x}^n$

where $m$ and $n$ are constants to be found.

Given that

$y=\frac{8}{27}{x}^6$

write

$y^{\frac{1}{3}}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Given that

$y=\frac{8}{27}{x}^6$

write

$y^{-1}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Given that

$y=\frac{8}{27}{x}^6$

write

$y^{-\frac{4}{3}}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

A teacher claims that

$\sqrt{a+b}=\sqrt{a}+\sqrt{b}$

for some values of $a$ and $b$.

Is the teacher correct? Show your reasoning clearly.

Show how to write

$\frac{3}{1+\sqrt{7}}$ 

in the form

$a+b\sqrt{7}$

without using a calculator, where $a$ and $b$ are rational numbers to be found.

Show how to write 

${256}^{\frac{1}{4}}$

as an integer, without using a calculator.

Use your answer to part (a) to find the value of 

$1÷{256}^{-\frac{3}{4}}$

without using a calculator.

Given that

$a^{\frac{2}{3}}=16$

find the possible values of $a$.

Write

$x^{-\frac{2}{3}}÷x^{-\frac{3}{4}}$

in the form

$x^{\frac{1}{n}}$

where $n$ is an integer to be found.

Solve the equation

$66+x\sqrt{75}=\frac{4x}{\sqrt{3}}$

giving your answer in the form

$x=a\sqrt{b}$

where $a$ and $b$ are integers to be found and your answer is in simplest form.

You must show the steps in your working clearly.

A trapezium has the following properties:

• The two parallel sides have lengths $\left(4+\sqrt{3}\right)$ cm and $\left(6-3\sqrt{3}\right)$ cm

• The area is $\left(10\sqrt{3}-6\right)$ cm²

Without using a calculator, find the perpendicular distance between the two parallel sides of the trapezium, showing the steps in your working clearly.

Give your answer in the form $a\sqrt{b}$ where $a$ and $b$ are integers to be found and your answer is in simplest form.

Use a non-calculator method to solve the equation

$\frac{729}{4\sqrt{3}-\sqrt{27}}=3^a$

to find the value of $a$.

Show the steps in your working clearly.

Expand and simplify

$\frac{{\left(5+2\sqrt{x}\right)}^2}{\sqrt{x}}$

giving each term in simplest form.

Show that

$\frac{2\sqrt{x}}{\sqrt{2}}\left(\sqrt{8x}+6x\right)$

can be written in the form

$2y^2+3y^3$

where $y$ is a function of $x$ which you should state.

Given that

$y=\frac{81}{16}{x}^{-12}$

write

$y^{\frac{3}{4}}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Given that

$y=\frac{81}{16}{x}^{-12}$

write

$y^{-\frac{1}{2}}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Given that

$y=\frac{81}{16}{x}^{-12}$

write

${\left(y^{\frac{1}{2}}\right)}^{-3}$

in the form $a{x}^n$ where $a$ and $n$ are constants to be found.

Write

${\left(\frac{2}{9}{x}^{\frac{1}{2}}\times \frac{1}{18}{x}^{-\frac{3}{4}}\right)}^{-\frac{1}{4}}$

in the form

$a{x}^{\frac{1}{n}}$

where $a$ and $n$ are integers to be found.

Write

${\left(8x^2\right)}^{-\frac{1}{3}}\times \frac{1}{4}{x}^{-\frac{1}{3}}$

in the form

$a{x}^k$

where $a$ and $k$ are constants to be found.

Show that

$\frac{{\left(8x^{-{\displaystyle \frac{2}{3}}}\right)}^{{\displaystyle \frac{2}{3}}}}{{\left(64x^{-{\displaystyle \frac{1}{3}}}\right)}^{{\displaystyle \frac{1}{3}}}}$

can be written in the form

$x^k$

where $k$ is a constant to be found.

A square has an area of $\left(49+12\sqrt{5}\right)$ m² and a side length of $\left(a+b\sqrt{5}\right)$  m, where $a$ and $b$ are rational numbers with $a>0$ and $b>0$.

Use algebra to find $a$ and $b$.