Worked examples · Sheet 1

Simplifications with the properties of powers

Theory used: Properties of powers Power with natural and integer exponent

Exercise 1

Simplify

\frac{2 ^{5} \cdot 2 ^{3}}{2 ^{6}} .

\frac{2 ^{5} \cdot 2 ^{3}}{2 ^{6}} = \frac{2 ^{5 + 3}}{2 ^{6}} = 2^{8 - 6} = 2^{2} = 4

In the numerator the two powers have the same base, therefore the product is the power with the sum of the exponents, that is, $2^{8}$ . The quotient of powers with the same base is the power with the difference of the exponents, hence the expression equals $2^{2} = 4$ .

Exercise 2

Simplify

\frac{6 ^{4} \cdot 2 ^{- 3}}{3 ^{4}} .

\frac{6 ^{4} \cdot 2 ^{- 3}}{3 ^{4}} = \frac{6 ^{4}}{3 ^{4}} \cdot 2^{- 3} = (\frac{6}{3})^{4} \cdot 2^{- 3} = 2^{4} \cdot 2^{- 3} = 2^{4 - 3} = 2

The powers $6^{4}$ and $3^{4}$ have the same exponent, therefore their quotient is the power of the quotient of the bases, that is, $(\frac{6}{3})^{4} = 2^{4}$ . At this point the product $2^{4} \cdot 2^{- 3}$ has a common base, hence the exponents are added and the result is $2^{1} = 2$ .

The negative exponent indicates the reciprocal: $2^{- 3} = \frac{1}{8}$ , and the direct check confirms the result, because $2^{4} \cdot \frac{1}{8} = \frac{16}{8} = 2$ .

Exercise 3

Simplify, for $a, b \neq = 0$ ,

(\frac{a ^{3} b ^{2}}{a b ^{4}})^{2} \cdot \frac{b ^{5}}{a ^{4}} .

(\frac{a ^{3} b ^{2}}{a b ^{4}})^{2} \cdot \frac{b ^{5}}{a ^{4}} = (a^{3 - 1} b^{2 - 4})^{2} \cdot \frac{b ^{5}}{a ^{4}} = (a^{2} b^{- 2})^{2} \cdot \frac{b ^{5}}{a ^{4}} = a^{4} b^{- 4} \cdot \frac{b ^{5}}{a ^{4}} = a^{4 - 4} b^{- 4 + 5} = a^{0} b^{1} = b

Inside the fraction one applies the quotient of powers with the same base, separately for $a$ and for $b$ : the exponents are subtracted and $a^{2}$ and $b^{- 2}$ remain. The power of a power multiplies the exponents, therefore the square of the simplified fraction is $a^{4} b^{- 4}$ .

In the final product the powers of $a$ and those of $b$ are combined by adding the exponents: for $a$ one obtains $4 - 4 = 0$ and for $b$ one obtains $- 4 + 5 = 1$ . Since $a \neq = 0$ , $a^{0} = 1$ holds, hence the expression reduces to $b$ .