Algebraic Expansion with Exponents (Indices)

Algebraic Expansion with Exponents (Indices)

$\textit{Algebraic Expansion with Exponents}$

Expansion of algebraic expressions like $x^{\frac{1}{3}}(4x^{\frac{4}{5}}-3x^{\frac{3}{2}})$, $(4x^5 + 6)(5^x-7)$ and $(4^x + 7)^2$ are handled in the same way, using the same expansion laws to simplify expressions containing exponents:

$$ \large \begin{align} \displaystyle
a(a+b) &= ab+ac \\
(a+b)(c+d) &= ac+ad+bc+bd \\
(a+b)(a-b) &= a^2-b^2 \\
(a+b)^2 &= a^2 + 2ab + b^2 \\
(a-b)^2 &= a^2-2ab + b^2
\end{align} $$

Example 1

Expand and simplify $x^{-\frac{1}{2}}\left(x^{\frac{3}{2}} + 2x^{\frac{1}{2}}-3x^{-\frac{1}{2}}\right)$.

\( \begin{align} \displaystyle
&= x^{-\frac{1}{2}} \times x^{\frac{3}{2}} + x^{-\frac{1}{2}} \times 2x^{\frac{1}{2}}-x^{-\frac{1}{2}} \times 3x^{-\frac{1}{2}} \\
&= x^{-\frac{1}{2} + \frac{3}{2}} + 2x^{-\frac{1}{2} + \frac{1}{2}} -3x^{-\frac{1}{2}-\frac{1}{2}} \\
&= x^1 + 2x^0-3x^{-1} \\
&= x + 2-\dfrac{3}{x}
\end{align} \)

Example 2

Expand and simplify $(2^x + 1)(2^x-2)$.

\( \begin{align} \displaystyle
&= 2^x \times 2^x-2^x \times 2 + 2^x-2 \\
&= 2^{2x}-2^{x+1} + 2^x-2
\end{align} \)

Example 3

Expand and simplify $(4^x + 2)(8^x-2)$.

\( \begin{align} \displaystyle
&= 4^x \times 8^x-4^x \times 2 + 2 \times 8^x-4 \\
&= (2^2)^x \times (2^3)^x-(2^2)^x \times 2 + 2 \times (2^3)^x-4 \\
&= 2^{2x} \times 2^{3x}-2^{2x} \times 2 + 2 \times 2^{3x}-4 \\
&= 2^{5x}-2^{2x+1} + 2^{3x+1}-4
\end{align} \)

Example 4

Expand and simplify $(5^x-5^{-x})^2$.

\( \begin{align} \displaystyle
&= (5^x)^2-2 \times 5^x \times 5^{-x} + (5^{-x})^2 \\
&= 5^{2x}-2 \times 5^0 + 5^{-2x} \\
&= 5^{2x}-2 + \dfrac{1}{5^{2x}}
\end{align} \)

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Related Articles

Responses

Your email address will not be published. Required fields are marked *