Fractional index

Let's go through these math concepts from your notes.

Fractional Indices

The general rule for fractional indices is that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. The denominator of the fraction indicates the root, and the numerator indicates the power.

• $a^{\frac{1}{2}} = \sqrt{a}$
• $b^{\frac{1}{3}} = \sqrt[3]{b}$
• For $5^{\frac{1}{3}}$, the correct form is: $$5^{\frac{1}{3}} = \boxed{\sqrt[3]{5}}$$ (Your note incorrectly shows $\sqrt[3]{6}$.)
• $x^{\frac{2}{3}} = \sqrt[3]{x^2}$
• $p^{\frac{2}{3}} = \sqrt[3]{p^2}$

Powers of Fractions

When a fraction is raised to a power, both the numerator and the denominator are raised to that power. The general rule is $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

• For $\left(\frac{a}{b}\right)^3$, the correct form is: $$\left(\frac{a}{b}\right)^3 = \boxed{\frac{a^3}{b^3}}$$ (Your note incorrectly shows $\frac{a^2}{b}$.)
• $\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}$

Negative Indices

A negative index indicates the reciprocal of the base raised to the positive power. For a fraction, this means you invert the fraction and change the sign of the exponent. The general rule is $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$.

• For $\left(\frac{a}{b}\right)^{-1}$, the correct form is: $$\left(\frac{a}{b}\right)^{-1} = \boxed{\frac{b}{a}}$$

Pythagoras Theorem

Pythagoras Theorem applies to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

• If $a$ and $b$ are the lengths of the two shorter sides (legs) and $c$ is the length of the hypotenuse, then the theorem is: $$\boxed{a^2 + b^2 = c^2}$$