Laws of Exponents (Index Laws)

$\textbf{Laws of Exponents (Index Laws)}$

$a^x \times a^y = a^{x+y}$
To $\textit{multiply}$ numbers with the $\textit{same base}$, keep the base and $\textit{add}$ the exponents.

$\dfrac{a^x}{a^y} = a^x \div a^y = a^{x-y}$
To $\textit{divide}$ numbers with the $\textit{same base}$, keep the base and $\textit{substract}$ the exponents.

$(a^x)^y = a^{x \times y}$
When $\textit{raising a power to a power}$, keep the base and $\textit{multiply}$ the exponents.

$(ab)^x = a^xb^x$
The power of a product is the product of the powers.

$\Big(\dfrac{a}{b}\Big)^x = \dfrac{a^x}{b^y}, b \ne 0$
The power of a quotient is the quotient of the powers.

$a^0 = 1, a \ne 0$
Any non-zero number raised to the power of zero is 1.

$a^{-x} = \dfrac{1}{a^x}$ and $\dfrac{1}{a^{-x}} = a^x, a \ne 0$

Example 1

Simplify $a^5 \times a^6$ using Laws of Exponents (Index Laws).

\( \begin{align} \displaystyle
a^5 \times a^6 &= a^{5+6} \\
&= a^{11} \\
\end{align} \)

Example 2

Simplify $\dfrac{a^7}{a^3}$.

\( \begin{align} \displaystyle
\dfrac{a^7}{a^3} &= a^{7-3} \\
&= a^4 \\
\end{align} \)

Example 3

Simplify $(a^3)^4$.

\( \begin{align} \displaystyle
(a^3)^4 &= a^{3 \times 4} \\
&= a^{12} \\
\end{align} \)

Example 4

Simplify $(a^2b^3)^4$.

\( \begin{align} \displaystyle
(a^2b^3)^4 &= a^{2 \times 4} b^{3 \times 4} \\
&= a^8 b^{12}
\end{align} \)

Example 5

Write $\dfrac{a^{-2}}{b^{-3}}$ without negative exponents.

\( \begin{align} \displaystyle
\dfrac{a^{-2}}{b^{-3}} &= \dfrac{b^3}{a^2} \\
\end{align} \)


Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials 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 Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

Your email address will not be published.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