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Negative and fractional indices
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Foundational skills
Integers to the power of -1
\[ 5^{-1} = \frac{1}{5} \]
Write an integer to the power of -1 as a fraction.
Unit fractions to the power of -1
\[ \left(\frac{1}{3}\right)^{-1} = 3 \]
Evaluate a unit fraction raised to the power of -1.
Non-unit fractions to the power of -1
\[ \left(\frac{4}{5}\right)^{-1} = \frac{5}{4} \]
Write a fraction to the power of -1.
Negative indices (numerical)
Understanding negative indices
\[ 5^{-2} \rightarrow \frac{1}{25} \]
Convert a negative power to a fraction with a positive power.
Evaluating negative powers
\[ 2^{-3} = \frac{1}{8} \]
Find the numerical value of a negative power.
Converting fractions to negative powers
\[ \frac{1}{64} \rightarrow 2^{-6} \]
Express a unit fraction as a base raised to a negative power.
Negative powers of fractions
\[ \left(\frac{1}{3}\right)^{-2} = 9 \]
Find the value of a fraction raised to a negative power.
Evaluating 10 to negative powers
\[ 10^{-2} = 0.01 \]
Calculate 10 raised to a negative power.
Negative indices (algebraic)
Algebraic terms with negative indices
\[ x^{-3} = \frac{1}{x^3} \]
Rewrite an algebraic negative power as a fraction with positive power.
Rewriting fractions as negative powers
\[ \frac{1}{a^4} = a^{-4} \]
Express a reciprocal of an algebraic power using a negative index.
Terms with coefficients and negative indices
\[ 3x^{-2} = \frac{3}{x^2} \]
Rewrite a term with a coefficient and negative index.
Division law resulting in negative indices
\[ x^3 \div x^7 = x^{-4} \]
Apply the division law when the result has a negative index.
Fractional indices (unit fractions)
Understanding the half power
\[ 16^{\frac{1}{2}} \rightarrow \sqrt{16} \]
Know that raising to the power 1/2 is the same as square root.
Evaluate a^(1/2)
\[ 49^{\frac{1}{2}} = 7 \]
Calculate a number raised to the power 1/2.
Understanding the third power
\[ 8^{\frac{1}{3}} \rightarrow \sqrt[3]{8} \]
Know that raising to the power 1/3 is the same as cube root.
Evaluate a^(1/3)
\[ 125^{\frac{1}{3}} = 5 \]
Calculate a number raised to the power 1/3.
Understanding and evaluating a^(1/4)
\[ 81^{\frac{1}{4}} = 3 \]
Know that raising to the power 1/4 is the same as fourth root and evaluate.
Evaluate a^(1/5)
\[ 32^{\frac{1}{5}} = 2 \]
Calculate a number raised to the power 1/5.
Convert between root and index notation
\[ \sqrt[3]{64} = 64^{\frac{1}{3}} \]
Convert a root expression to index notation and vice versa.
Fractional indices (general)
Understand the meaning of a^(m/n)
\[ 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 \]
Know that a^(m/n) means the nth root of a raised to the power m.
Evaluate a^(3/2)
\[ 16^{\frac{3}{2}} = 64 \]
Calculate a number raised to the power 3/2.
Evaluate a^(2/3)
\[ 27^{\frac{2}{3}} = 9 \]
Calculate a number raised to the power 2/3.
Evaluate a^(4/3)
\[ 8^{\frac{4}{3}} = 16 \]
Calculate a number raised to the power 4/3.
Evaluate a^(3/4)
\[ 16^{\frac{3}{4}} = 8 \]
Calculate a number raised to the power 3/4.
Evaluate other fractional powers
\[ 4^{\frac{5}{2}} = 32 \]
Calculate a number raised to powers like 5/2 or 2/5.
Negative fractional indices
Understand negative fractional indices
\[ 16^{-\frac{1}{2}} = \frac{1}{4} \]
Know that a^(-1/n) means 1 divided by the nth root of a.
Evaluate a^(-1/2)
\[ 25^{-\frac{1}{2}} = \frac{1}{5} \]
Calculate a number raised to the power -1/2.
Evaluate a^(-1/3) and a^(-1/4)
\[ 8^{-\frac{1}{3}} = \frac{1}{2} \]
Calculate a number raised to the power -1/3 or -1/4.
Evaluate a^(-3/2)
\[ 4^{-\frac{3}{2}} = \frac{1}{8} \]
Calculate a number raised to the power -3/2.
Evaluate a^(-2/3)
\[ 27^{-\frac{2}{3}} = \frac{1}{9} \]
Calculate a number raised to the power -2/3.
Evaluate other negative fractional indices
\[ 8^{-\frac{4}{3}} = \frac{1}{16} \]
Calculate a number raised to powers like -4/3 or -3/4.
Rewrite with positive indices
\[ 49^{-\frac{3}{2}} \rightarrow \frac{1}{343} \]
Rewrite a negative fractional index as a fraction with a positive index.
Algebraic applications
Algebraic terms with half power
\[ x^{\frac{1}{2}} \times x^{\frac{1}{2}} = x \]
Work with algebraic terms raised to the power of one half.
Algebraic terms with fractional indices
\[ a^{\frac{2}{3}} \times a^{\frac{1}{3}} = a \]
Multiply algebraic terms with fractional indices.
Rewriting roots as fractional powers
\[ \sqrt{x} = x^{\frac{1}{2}} \]
Express a root using fractional index notation.
Rewriting fractional powers as roots
\[ a^{\frac{3}{4}} = \sqrt[4]{a^3} \]
Express a fractional power using root notation.
Division with fractional indices
\[ x^{\frac{5}{6}} \div x^{\frac{1}{6}} = x^{\frac{2}{3}} \]
Divide algebraic terms with fractional indices by subtracting.
Combined skills
Combining positive and negative indices
\[ x^5 \times x^{-2} \times x^3 \]
Combine terms including negative indices using addition of indices.
Combining fractional and integer indices
\[ a^{\frac{1}{2}} \times a^{\frac{3}{2}} \times a^2 \]
Combine fractional and integer indices.
Find the missing index
\[ 8^x = 4 \rightarrow x = \frac{2}{3} \]
Find the value of an unknown index where the answer is a fraction.
Compare expressions without full calculation
\[ 8^{\frac{2}{3}} \text{ vs } 8^{\frac{3}{4}} \]
Compare two expressions with the same base but different fractional indices.
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