Operation on Exponent Numbers

Multiplication in Exponent Numbers

Multiplication of of exponent numbers that have the same base/principal number is formulated as follows:

\[\boxed{{a^m} \times {a^n} = {a^{m + n}}}\]

Example:

1. ${2^3} \times {2^4} = {2^{3 + 4}} = {2^7}$
2. ${\left( { - 3} \right)^3} \times {\left( { - 3} \right)^2} = {\left( { - 3} \right)^{3 + 2}} = {\left( { - 3} \right)^5}$
3. ${p^2} \times {p^5} = {p^{2 + 5}} = {p^7}$
4. ${\left( {\frac{1}{2}} \right)^3} \times {\left( {\frac{1}{2}} \right)^2} = {\left( {\frac{1}{2}} \right)^{3 + 2}} = {\left( {\frac{1}{2}} \right)^5}$
5. ${3^2} \times {3^3} \times {3^4} = {3^{2 + 3 + 4}} = {3^9}$

Dividing of exponent numbers

The division of exponent numbers that have the same base/base is formulated as follows:

\[\boxed{{a^m} \div {a^n} = {a^{m - n}}}\]

Example:

1. ${2^4} \div {2^2} = {2^{4 - 2}} = {2^2}$
2. ${\left( { - 3} \right)^3} \div {\left( { - 3} \right)^2} = {\left( { - 3} \right)^{3 - 2}} = {\left( { - 3} \right)^1} = \left( { - 3} \right)$
3. ${q^5} \div {q^2} = {q^{5 - 2}} = {q^3}$
4. ${\left( {\frac{1}{2}} \right)^3} \div {\left( {\frac{1}{2}} \right)^2} = {\left( {\frac{1}{2}} \right)^{3 - 2}} = {\left( {\frac{1}{2}} \right)^1} = \frac{1}{2}$
5. ${3^6} \div {3^2} \div {3^3} = {3^{6 - 2 - 3}} = {3^1} = 3$

Power of Exponent Numbers

The power of exponent numbers is formulated as follows:

\[\boxed{{\left( {{a^m}} \right)^n} = {a^{m \times n}}}\]

Example:

1. ${\left( {{4^2}} \right)^3} = {4^{2 \times 3}} = {4^6}$
2. ${\left( {{2^3}} \right)^5} = {2^{3 \times 5}} = {2^{15}}$
3. ${\left( {{{\left( { - 3} \right)}^2}} \right)^4} = {\left( { - 3} \right)^{2 \times 4}} = {\left( { - 3} \right)^8}$

Exponent of Multiplication of Numbers

The power of multiplication of numbers is formulated as follows:

\[\boxed{{\left( {a \times b} \right)^m} = {a^m} \times {b^m}}\]

Example:

1. ${\left( {2 \times 3} \right)^4} = {2^4} \times {3^4}$
2. ${\left( {2 \times 5} \right)^2} = {2^2} \times {5^2}$
3. ${\left( {3y} \right)^3} = {\left( {3 \times y} \right)^3} = {3^3} \times {y^3} = 27 \times {y^3} = 27{y^3}$
4. ${\left( {2pq} \right)^2} = {\left( {2 \times p \times q} \right)^2} = {2^2} \times {p^2} \times {q^2} = 4 \times {p^2} \times {q^2} = 4{p^2}{q^2}$

Exponent of Fractions

The power of fractions is formulated as follows:

\[\boxed{{\left( {\frac{a}{b}} \right)^m} = \frac{{{a^m}}}{{{b^m}}}}\]

Example:

1. ${\left( {\frac{2}{3}} \right)^2} = \frac{{{2^2}}}{{{3^2}}} = \frac{4}{9}$
2. ${\left( { - \frac{1}{2}} \right)^3} = {\left( {\frac{{ - 1}}{2}} \right)^3} = \frac{{{{\left( { - 1} \right)}^3}}}{{{2^3}}} = \frac{{ - 1}}{8} = - \frac{1}{8}$
3. ${\left( {2\frac{1}{3}} \right)^3} = {\left( {\frac{7}{3}} \right)^3} = \frac{{{7^3}}}{{{3^3}}} = \frac{{343}}{{27}} = 12\frac{{19}}{{27}}$

Examples of Operations with Exponent Numbers

1. Problem A
$\begin{array}{l} \frac{{{2^4} \times {2^2}}}{{{2^3}}} &= \frac{{{2^{4 + 2}}}}{{{2^3}}}\\ &= \frac{{{2^6}}}{{{2^3}}}\\ &= {2^{6 - 3}}\\ &= \boxed{{2^3}} \end{array}$
2. Problem B
$\begin{array}{l} {3^3} \times {9^2} \times {27^2} &= {3^3} \times {\left( {{3^2}} \right)^2} \times {\left( {{3^3}} \right)^2}\\ &= {3^3} \times {3^{2 \times 2}} \times {3^{3 \times 2}}\\ &= {3^3} \times {3^4} \times {3^6}\\ &= {3^{3 + 4 + 6}}\\ &= \boxed{{3^{13}}} \end{array}$
3. Problem C
$\begin{array}{l} {5^2} \times {\left( {\frac{2}{5}} \right)^3} \times {\left( {\frac{2}{5}} \right)^5} &= {5^2} \times \frac{{{2^3}}}{{{5^3}}} \times \frac{{{2^5}}}{{{5^5}}}\\ &= \frac{{{5^2} \times {2^3} \times {2^5}}}{{{5^3} \times {5^5}}}\\ &= \frac{{{5^2} \times {2^{3 + 5}}}}{{{5^{3 + 5}}}}\\ &= \frac{{{5^2} \times {2^8}}}{{{5^8}}}\\ &= \frac{{{5^2} \times {2^8}}}{{{5^8}}} \div \frac{{{5^2}}}{{{5^2}}}\\ &= \frac{{{5^{2 - 2}} \times {2^8}}}{{{5^{8 - 2}}}}\\ &= \frac{{{5^0} \times {2^8}}}{{{5^6}}}\\ &= \frac{{1 \times {2^8}}}{{{5^6}}}\\ &= \boxed{\frac{{{2^8}}}{{{5^6}}}} \end{array}$
4. Problem D
$\begin{array}{l} b \times 2{y^7} \times {b^3} \times {y^2} &= \underline {b \times {b^3}} \times \underline {2{y^7} \times {y^2}} \\ &= {b^{1 + 3}} \times 2 \times {y^{7 + 2}}\\ &= {b^4} \times 2 \times {y^9}\\ &= 2 \times {b^4} \times {y^9}\\ &= \boxed{2{b^4}{y^9}} \end{array}$

That's the material and examples of operations on exponent numbers. Hope it's useful