The exponents and roots section features a laws of indices page.
Which mostly shows examples of how to deal with simplifying and calculating when it is numbers that are affected by exponents.
But it’s the case that Mathematical expressions and equations can also include variables affected by exponents, and these expressions can also be simplified and manipulated sometimes when needed.
Here we will see examples of simplifying expressions with exponents present.
Examples
1.1
a2 × a5
This can be simplified by first considering each term separately.
a2 = Two a‘s multiplied together. a × a
a5 = Five a‘s multiplied together. a × a × a × a × a
So: a2 × a5 = Seven a‘s multiplied together. a × a × a × a × a × a × a
Thus: a2 × a5 = a7
Generally: am × an = am + n
1.2
(b3 )2
This expression can be simplified by looking at what’s inside the brackets, and looking at what effect the exponent outside the brackets has.
(b3 )2 = Three b‘s multiplied together, twice.
b3 = b × b × b
(b3 )2 = ( b × b × b ) × ( b × b × b ) = b × b × b × b × b × b × b = b6
Generally: (bm )n = bm × n
1.3
a) \bf{\frac{b^5}{b^3}}
We can attempt simplifying expressions with exponents like this by thinking about what each term on the top and bottom of the fraction represent.
\bf{\frac{b^5}{b^2}} = \bf{\frac{b \times b \times b \times b \times b}{b \times b}}
Some of the b‘s can be cancelled out with each other.
\bf{\frac{\cancel{b} \times \cancel{b} \times b \times b \times b}{\cancel{b} \times \cancel{b}}} = \bf{\frac{b \times b \times b}{1}} = \bf{b \times b \times b} = \bf{b^3}
\bf{\frac{b^5}{b^2}} = \bf{b^{5-2}}
b) \bf{\frac{b^2}{b^5}}
Can again simplify by using the same thought process.
\bf{\frac{b^2}{b^5}} = \bf{\frac{b \times b}{b \times b \times b \times b \times b}}
=> \bf{\frac{\cancel{b} \times \cancel{b}}{\cancel{b} \times \cancel{b} \times b \times b \times b}} = \bf{\frac{1}{b \times b \times b}} = \bf{\frac{1}{b^3}} = \bf{b^{-3}}
\bf{\frac{b^2}{b^5}} = \bf{b^{2-5}}
Generally: \bf\color{darkred}{\frac{b^m}{b^n}} = \bf\color{darkred}{b^{m - n}}
1.4
a) Simplify \bf{\frac{2a^4}{3a^2}}.
Solution
\bf{\frac{2a^4}{3a^2}} = \bf{\frac{2}{3}} × \bf{\frac{a^4}{a^2}} = \bf{\frac{2}{3}} × \bf{a^2} = \bf{\frac{2a^2}{3}}
b) Simplify \bf{\frac{7a^5}{4a^8}}.
Solution
\bf{\frac{7a^5}{4a^8}} = \bf{\frac{7}{5}} × \bf{\frac{a^5}{a^8}} = \bf{\frac{7}{4}} × \bf{a^{-3}} = \bf{\frac{7}{4}} × \bf{\frac{1}{a^3}} = \bf{\frac{7}{4a^3}}
(1.5)
a) Simplify a−4 × a2.
Solution
a-4 × a2 = a−4 + 2 = a−2 = \bf{\frac{1}{a^2}}
b) Simplify (b2 )3.
Solution
(b2 )3 = b2 × 3 = b6
Simplifying Expressions with Exponents
Further Examples
2.1
a) Simplify 3a2b4 × 2ab2.
Solution
A good first step in simplifying expressions with exponents such as this, is to look to group like terms together, then proceed.
3 × 2 × a2a × b4b2 = 6 × a3 × b6 = 6a3b6
b) Simplify ( 2a3b2 )2.
Solution
( 2a3b2 )2 = 2a3b2 × 2a3b2
= 2 × 2 × a3 × a3 × b2b2 = 4 × a6 × b4 = 4a6b4
(2.2)
a) Simplify \bf{\frac{5a^2 b^3}{3a b^5}}.
Solution
Similar to before, look to group into separate fractions containing like terms.
\bf{\frac{5a^2 b^3}{3a b^5}} = \bf{\frac{5}{3}} × \bf{\frac{a^2}{a}} × \bf{\frac{b^3}{b^5}} = \bf{\frac{5}{3}} × \bf{a} × \bf{b^{-2}}
= \bf{\frac{5a}{3}} × \bf{\frac{1}{b^2}} = \bf{\frac{5a}{3b^2}}
b) Simplify \bf{\frac{7abc^4}{9b^2 c^3}}.
Solution
\bf{\frac{7}{9}} × \bf{a} × \bf{\frac{b}{b^2}} × \bf{\frac{c^4}{c^3}} = \bf{\frac{7a}{9}} × \bf{\frac{1}{b}} × \bf{c} = \bf{\frac{7a}{9}} × \bf{\frac{c}{b}} = \bf{\frac{7ac}{9b}}