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Index Laws
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Foundational skills
Index notation with algebraic terms
\[ a \times a \times a \times a \rightarrow a^4 \]
Write a repeated multiplication using index notation.
Expanding algebraic powers
\[ m^4 \rightarrow m \times m \times m \times m \]
Write an algebraic power as a repeated multiplication.
Multiplication law
Multiplying powers (Numerical)
\[ 3^4 \times 3^2 \rightarrow 3^6 \]
Simplify by adding indices.
Multiplying terms with coefficients
\[ 3a^4 \times 5a^2 \rightarrow 15a^6 \]
Multiply coefficients and add indices.
Multiplying multiple variables
\[ 2a^3b^2 \times 4a^2b^5 \rightarrow 8a^5b^7 \]
Multiply terms containing more than one variable.
Division law
Dividing powers (Numerical)
\[ 5^7 \div 5^3 \rightarrow 5^4 \]
Simplify by subtracting indices.
Dividing terms with coefficients
\[ 24m^7 \div 6m^3 \rightarrow 4m^4 \]
Divide coefficients and subtract indices.
Simplifying algebraic fractions
\[ \frac{15x^6}{3x^2} \rightarrow 5x^4 \]
Simplify fractions with algebraic terms.
Power of a power
Power of a power (Numerical)
\[ (4^3)^2 \rightarrow 4^6 \]
Simplify by multiplying indices.
Power of a power (Coefficients)
\[ (2a^3)^4 \rightarrow 16a^{12} \]
Raise both coefficient and variable to the power.
Zero index
Understanding the zero index
\[ 8^0 \rightarrow 1 \]
Evaluate terms with power zero.
Zero index in expressions
\[ 5a^0 + 3b^0 \rightarrow 8 \]
Simplify expressions containing zero powers.
Combined laws
Multiplication and division (Numerical)
\[ \frac{3^5 \times 3^2}{3^4} \rightarrow 3^3 \]
Simplify using both laws on numeric bases.
Multiplication and division (Algebra)
\[ \frac{x^5 \times x^3}{x^2} \rightarrow x^6 \]
Apply both multiplication and division laws.
Multiplication and division (Coeffs)
\[ \frac{4a^5 \times 6a^2}{3a^3} \rightarrow 8a^4 \]
Simplify terms with coefficients using both laws.
Power of a power with multiplication
\[ (a^3)^2 \times a^4 \rightarrow a^{10} \]
Combine power of a power with the multiplication law.
Special cases
Finding the unknown index
\[ 3^n \times 3^4 = 3^9 \rightarrow n=5 \]
Find an unknown index using the multiplication law.
Solving equations
\[ 2^{x+1} = 16 \rightarrow x=3 \]
Use index laws to solve equations involving powers.
Timer (Optional)
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