Algebraic Roots & Indices (Cambridge (CIE) O Level Maths): Revision Note

(a) Simplify ${\left(u^5\right)}^5$

Answer:

 Use ${\left(a^m\right)}^n=a^{mn}$

${\left(u^5\right)}^5=u^{5\times 5}$

$u^{25}$

(b) If  $q^x=\frac{q^2\times q^5}{q^{10}}$   find $x$.

Answer:

Use $a^m\times a^n=a^{m+n}$ to simplify the numerator

$q^2\times q^5=q^{2+5}=q^7$

Use $\frac{a^m}{a^n}=a^{m-n}$ to simplify the fraction

$\frac{q^7}{q^{10}}=q^{7-10}=q^{-3}$

Write out both sides of the equation

$q^x=q^{-3}$ 

Both sides are now over the same base of $q$

So $x$ must equal the power on the right-hand side

$x=-3$

(a) Rewrite $\frac{1}{\sqrt[3]{x^4}}$ in the form $x^n$ where $n$ is a negative fraction.

Answer:

Use $a^{\frac{1}{n}}=\sqrt[n]{a}$ to rewrite the cube-root as a power of $\frac{1}{3}$

$\frac{1}{{\left(x^4\right)}^{{\displaystyle \frac{1}{3}}}}$

Use ${\left(a^m\right)}^n=a^{mn}$ to simplify the denominator

$\frac{1}{x^{{\displaystyle \frac{4}{3}}}}$

Use $a^{-n}=\frac{1}{a^n}$ to rewrite as a term with a negative fraction as the power

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

(b) Find the value of the constants $m$ and $a$ given that ${\left(a{x}^6\right)}^{\frac{1}{m}}=8x^3$.

Answer:

Use ${\left(ab\right)}^n=a^n{b}^n$ to rewrite the left hand side
Remember to apply the power to both $a$ and $x^6$

$a^{\frac{1}{m}}\times x^{\frac{6}{m}}=8x^3$

Both sides of the equation have a constant part, $a^{\frac{1}{m}}$ and $8$
And both sides of the equation have a part in terms of $x$

The two sides of the equation are equal, so set the respective parts equal to one another

First,

$x^{\frac{6}{m}}=x^3$

The bases are the same, therefore the powers are equal

$\frac{6}{m}=3$

Solve to find $m$

$m=2$

Then set the constant parts of both sides equal to one another

$a^{\frac{1}{m}}=8$

We now know that $m=2$, so substitute this in

$a^{\frac{1}{2}}=8$

Use $a^{\frac{1}{n}}=\sqrt[n]{a}$ to rewrite as a square root

$\sqrt[2]{a}=8$

Find $a$ by squaring both sides

$a=64$