Laws of Indices & Surds (Edexcel International AS Maths: Pure 1): Exam Questions

Write down the value of ${27}^{\frac{1}{3}}$

Use your answer to part (a) to show that ${27}^{\frac{2}{3}}=9$.

Given that $a^{\frac{1}{3}}=2$, find the value of a.

Simplify $x^2÷x^{\frac{5}{3}}$

Simplify the following expressions:

$4x^2\times 3x^{-1}$

$24x^3÷8x^{\frac{11}{5}}$

$15x^{-\frac{2}{3}}÷10x^{-\frac{2}{3}}$

Given that $y=\frac{1}{16}{x}^4$, express each of the following in the form $a{x}^n$, where a and n are constants.

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

$y^{-1}$

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

Simplify $3\sqrt{5}+4\sqrt{3}-7\sqrt{3}+\sqrt{5}$

By expanding and simplifying, show that

$\left(\sqrt{3}-5\right)\left(\sqrt{3}+5\right)=-22$

Give an example to show that $\sqrt{a}+\sqrt{b}=\sqrt{a+b}$ is not true in general.

Show that $\frac{14}{4+\sqrt{2}}=4-\sqrt{2}$.

Solve the equation  $6-x\sqrt{7}=\frac{2x}{\sqrt{7}}$, giving your answer in the form $\frac{a\sqrt{7}}{b}$ where a and b are integers.

A rectangle has an area of 14 m² and a length of $\left(3-\sqrt{2}\right)$ m.  Find the width of the rectangle, showing clear algebraic working. Give your answer as an exact value.

Show that  $\frac{{\left(3-\sqrt{x}\right)}^2}{x}$ can be written as $9x^{-1}-6x^{-\frac{1}{2}}+1$.

Given that $128\sqrt{2}=2^a$, find the value of a.

Show that  $\frac{x\left(2x^4-\sqrt{x}\right)}{\sqrt{x}}$ can be written as $2x^a-x^b$, where a and b are rational numbers to be found.

Write down the value of ${64}^{\frac{1}{3}}$.

Use your answer to part (a) to show that ${64}^{-\frac{2}{3}}=\frac{1}{16}$

Given that $a^{\frac{1}{3}}=-4$, find the value of a.

Simplify $x^{\frac{11}{6}}÷x^{\frac{4}{3}}$

Simplify the following expressions, giving your answers in the form $a{x}^n$ where $a$ and $n$ are rational numbers and any fractions are in lowest terms.

$7x^{-3}÷2x^{-\frac{1}{3}}$

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

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

Given that $y=\frac{8}{27}{x}^6$, express each of the following in the form $a{x}^n$, where a and b are constants.

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

$y^{-1}$

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

Show that  $2\sqrt{3}+4\sqrt{12}-3\sqrt{75}=-5\sqrt{3}$

By expanding and simplifying, show that

$\left(\sqrt{3}-5\right)\left(5-\sqrt{3}\right)=10\sqrt{3}-28$

$\sqrt{a}+\sqrt{b}=\sqrt{a+b}$ is not true in general. Give an example of an a and a b for which it is true.

Show that $\frac{3}{1+\sqrt{7}}=a+b\sqrt{7}$ , where a and b are rational numbers to be found.

Solve the equation $66+x\sqrt{75}=\frac{4x}{\sqrt{3}}$, giving your answer in the form $a\sqrt{b}$ where a and b are integers.

The two parallel sides of a trapezium have lengths of $\left(4+\sqrt{3}\right)$ cm and $\left(6-3\sqrt{3}\right)$ cm, and the area of the trapezium is $\left(10\sqrt{3}-6\right)$ cm².   Showing clear algebraic working, determine the perpendicular distance between the two parallel sides of the trapezium.  Give your answer in the form $a\sqrt{b}$ where a and b are integers.

Show that $\frac{{\left(5+2\sqrt{x}\right)}^2}{\sqrt{x}}$  can be written as $25x^{-\frac{1}{2}}+20+4x^{\frac{1}{2}}$.

Show that $\frac{729}{4\sqrt{3}-\sqrt{27}}=3^a$, where a is a rational number to be found.

Show that  $\frac{2\sqrt{x}}{\sqrt{2}}\left(\sqrt{8x}+6x\right)$  can be written as $a{y}^2+b{y}^3$, where $y=\sqrt{2x}$ and a and b are integers to be found.

Write down the value of ${256}^{\frac{1}{4}}$

Use your answer to part (a) to show that $1÷{256}^{-\frac{3}{4}}=64$.

Given that $a^{\frac{2}{3}}=16$, find the possible values of a.

Simplify $x^{-\frac{2}{3}}÷x^{-\frac{3}{4}}$

Simplify the following expressions, giving your answers in the form $a{x}^n$ where a and n are rational numbers and any fractions are in lowest terms.

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

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

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

Given that $y=\frac{81}{16}{x}^{-12}$, express each of the following in the form $a{x}^n$, where a and n are constants.

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

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

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

Show that $2\sqrt{18}+\sqrt{50}-5\sqrt{32}=a\sqrt{b}$, where a and b are integers.

By expanding and simplifying, show that

$\left(\sqrt{12}-3\right)\left(2-\sqrt{75}\right)=19\sqrt{3}-36$

$\sqrt{a}-\sqrt{b}=\sqrt{a-b}$ is not true in general. Give an example of an a and a b for which it is true.

Show that  $\frac{2-\sqrt{3}}{1+\sqrt{3}}=a+b\sqrt{3}$ , where a and b are rational numbers.

Solve the equation  $\sqrt{20}+\frac{\sqrt{5}}{2x}=\frac{1}{x\sqrt{45}}$

Expand ${\left(a+b\sqrt{5}\right)}^2$.

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.

Show that $ab=6$, and explain why this proves that a and b must both be non-negative.

Show that $a^4-49a^2+180=0$.

By using the substitution $y=a^2$ or otherwise, solve the equation $a^4-49a^2+180=0$ .  Hence determine the side length of the square.