Indicies

The Laws and applications of Indices including Surds

Simplify $6x^{a}\times3x^{b}$

$$18x^{a+b}$$

Simplify $6x^{a}$ ÷ $3x^{b}$

$$2x^{a-b}$$

Simplify $\frac{9x^{a}}{3x^{b}}$

$$3x^{a-b}$$

Simplify $\left(3x^{a}\right)^{b}$

$$3^{b}x^{a\times b}$$

Rationalise the denominator of

$$\frac{3}{\sqrt5}$$

$$\frac{3}{\sqrt5}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{3\sqrt5}{5}$$

Rationalise the denominator of

$$\frac{3}{4-\sqrt2}$$

$$\frac{3}{4-\sqrt2}\times\frac{4+\sqrt2}{4+\sqrt2}=\frac{12+3\sqrt2}{16+4\sqrt2-4\sqrt2-2}=\frac{12+3\sqrt2}{14}$$

Simplify $x^0$

1

Simplify $25^{\frac12}$

$$25^{\frac12}=\sqrt{25}=5$$

Simplify $4^{-3}$

$$\frac{1}{4^3}^{}=\frac{1}{4\times4\times4}=\frac{1}{64}$$

Simplify $27^{\frac{^2}{3}}$

$$\left(27^{\frac13}\right)^2=\left(3\right)^2=9$$

Simplify $\left(\frac34\right)^{^{-1}}$

$$4/3$$

Simplify $\sqrt{5}\times\sqrt{3}$

$$\sqrt{15}$$

Simplify $\sqrt{75}$

$$5\sqrt3$$

Solve $\frac{x\times\sqrt5}{\sqrt{50}}=\sqrt3$

$$\sqrt{30}$$

Simplify $5\sqrt{27}-2\sqrt{75}$

$$5\sqrt3$$

$$\left(2^4\right)^{\frac15}\times2^{x}=\left(2^3\right)^{\frac34}\rightarrow2^{\frac45}\times2^{x}=2^{\frac94}\rightarrow\frac45+x=\frac94\rightarrow x=\frac94-\frac45=\frac{45}{20}-\frac{16}{20}=\frac{29}{20}$$