Indices and surds – advanced

This lesson comprises two (2) master classes focusing on:

• Index laws
• The zero index and power of a power
• Negative indices
• Fractional indices and surds
• Operations with surds

Content:

MA5-IND-P-01

Apply index laws to algebraic expressions involving negative-integer indices

• Apply index notation, patterns and index laws to establish \( a^{-1}=\frac{1}{a} \), \( a^{-2}=\frac{1}{a^2} \), \( a^{-3}=\frac{1}{a^3} \) and \( a^{-n}=\frac{1}{a^n} \)
• Represent expressions involving negative-integer indices as expressions involving positive-integer indices and vice versa
• Apply the index laws to simplify algebraic products and quotients involving negative-integer indices
• Describe and use \(x^{-1}\) as the reciprocal of \( x \) and generalise this relationship to expressions of the form \( \ (\frac{a}{b})^{-1} \)
• Use knowledge of the reciprocal to simplify expressions of the form \( \ (\frac{a}{b})^{-n} \)

MA5-IND-P-02

Describe surds

• Describe a real number as a number that can be represented by a point on the number line
• Examine the differences between rational and irrational numbers and recognise that all rational and irrational numbers are real
• Convert between recurring decimals and their fractional form using digital tools
• Describe the term surd as referring to irrational expressions of the form (\( \sqrt[n]{x} \)) where \( x \) is a rational number and \( n \) is an integer such that \( n \ge 2 \), and \( x>0 \) when \( n \) is even
• Recognise that a surd is an exact value that can be approximated by a rounded decimal
• Demonstrate that \( \sqrt{x} \) is undefined for \( x<0 \) and that \( \sqrt{x}=0 \) when \( x=0 \) using digital tools
• Describe (\( \sqrt{x} \)) as the positive square root of \( x \) for \( x>0 \) and \( \sqrt{0}=0 \)

Apply knowledge of surds to solve problems

• Establish and apply the following results for \( x>0 \) and \( y>0 \): \( (\sqrt{x})^2=x=\sqrt{x^2} \),  \( \sqrt{xy}=\sqrt{x} \times \sqrt{y} \) and  \( \sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}} \)
• Apply the 4 operations to simplify expressions involving surds
• Expand and simplify expressions involving surds
• Rationalise the denominators of surds of the form \( \frac{a \sqrt{b}}{c \sqrt{d}} \)

Describe and use fractional indices

• Apply index laws to describe fractional indices as: \( a^{\frac{1}{n}} =\sqrt[n]{a} \) and \( a^{\frac{m}{n}} =\sqrt[n]{a^m}=(\sqrt[n]{a})^m \)
• Translate expressions in surd form to expressions in index form and vice versa
• Evaluate numerical expressions involving fractional indices, including using digital tools