Table of Contents
Last modified on October 26th, 2024
Multiplying exponents is when two terms with exponents (also called indices or powers) are multiplied. However, dividing exponents is when two exponential terms are divided.
There are two possible cases:
When the terms with the same base are multiplied, the powers are added. Let us multiply two terms, xm and xn. The product is xm × xn = xm + n
For example,
25 × 27 = 25 + 7 = 212
E.g.1.
Multiply 53 × 52
Here, 53 × 52
= 53 + 2
= 55
Problem: Multiplying expressions with VARIABLES
Multiply a4 × a5
Here, a4 × a5
= a4 + 5
= a9
When the terms with different bases and the same exponents are multiplied, we multiply the bases by keeping the exponent the same over the product. Let us multiply xm and ym. The product is xm × ym = (xy)m
For example,
25 × 35 = (2 × 3)5 = 65
Multiply 53 × 73
Here, 53 × 73
= (5 × 7)3
= 353
Problem: Multiplying expressions with VARIABLES
Multiply a8 × b8
Here, a8 × b8
= (a × b)8
= (ab)8
When the expressions with different bases and different powers are multiplied, we evaluate each expression separately and then multiply.
Let us multiply xm and yn. The product is xm × yn = xmyn
Multiply 103 × 32 × 71
Here, 103 × 32 × 71
= 1000 × 9 × 7
= 63000
Negative Exponents represent how many times we need to multiply the reciprocal of the base. In other words, a negative exponent can be converted to a positive one by considering the reciprocal of the given term, after which it can be solved just like a positive exponent.
When multiplying negative exponents, we follow the rules given below.
Multiply 3-8 × 9-2 × 3-1
Here, 3-8 × 9-2 × 3-1
= 3-8 × (32)-2 × 3-1
= 3-8 × 3-4 × 3-1
= 3-8 + (-4) + (-1)
= 3-13
When multiplying terms with fractional exponents, we apply the following exponent rules.
Multiply ${4^{\dfrac{2}{3}}\times 2^{\dfrac{3}{2}}}$
Here, ${4^{\dfrac{2}{3}}\times 2^{\dfrac{3}{2}}}$
= ${\left( 2^{2}\right) ^{\dfrac{2}{3}}\times 2^{\dfrac{3}{2}}}$
= ${2^{2\times \dfrac{2}{3}}\times 2^{\dfrac{3}{2}}}$
= ${2^{\dfrac{4}{3}}\times 2^{\dfrac{3}{2}}}$
= ${2^{\dfrac{4}{3}+\dfrac{3}{2}}}$
= ${2^{\dfrac{8+9}{6}}$
= ${2^{\dfrac{17}{6}}$
Here is a summary of the laws we follow while multiplying expressions with exponents:
Like multiplying exponents, the division of exponents can have two possible cases:
When the terms with the same base are multiplied, the power of the divisor is subtracted from the power of the dividend.
Let us divide xm by xn. The quotient obtained is xm ÷ xn = ${\dfrac{x^{m}}{x^{n}}=x^{m-n}}$
For example,
${\dfrac{7^{5}}{7^{3}}=7^{5-3}=7^{2}}$
Evaluate ${\dfrac{4^{5}\times 3^{3}}{4^{2}\times 3^{2}}}$
Here, ${\dfrac{4^{5}\times 3^{3}}{4^{2}\times 3^{2}}}$
= ${\left( \dfrac{4^{5}}{4^{2}}\right) \times \left( \dfrac{3^{3}}{3^{2}}\right)}$
= ${\left( 4^{5-2}\right) \times \left( 3^{3-2}\right)}$
= 43 × 31
= 64 × 3
= 192
There are two possible cases:
When the terms with different bases and the same exponents are divided, we divide the bases by keeping the exponent the same over the quotient.
Let us divide xm by ym. The quotient is xm ÷ ym = ${\dfrac{x^{m}}{y^{m}}=\left( \dfrac{x}{y}\right) ^{m}}$
For example,
${\dfrac{7^{3}}{5^{3}}=\left( \dfrac{7}{5}\right) ^{3}}$
Divide ${\dfrac{4^{5}}{2^{5}}}$
Here, ${\dfrac{4^{5}}{2^{5}}}$
= ${\left( \dfrac{4}{2}\right) ^{5}}$
= 25
When the expressions with different bases and different powers are divided, we evaluate each expression separately and then divide.
For example,
${\dfrac{2^{5}}{3^{2}}}$ = ${\dfrac{32}{9}}$
Divide ${\dfrac{3^{3}}{10^{2}}}$
Here, ${\dfrac{3^{3}}{10^{2}}}$
= ${\dfrac{27}{100}}$
= 0.27
When dividing negative exponents, we follow the following rules.
When multiplying terms with fractional exponents, we apply similar exponent rules.
Divide ${4^{\dfrac{2}{3}}\div 2^{\dfrac{1}{2}}}$
Here, ${4^{\dfrac{2}{3}}\div 2^{\dfrac{1}{2}}}$
= ${\dfrac{4^{\dfrac{2}{3}}}{2^{\dfrac{1}{2}}}}$
= ${\dfrac{\left( 2^{2}\right) ^{\dfrac{2}{3}}}{2^{\dfrac{1}{2}}}}$
= ${\dfrac{2^{2\times \dfrac{2}{3}}}{2^{\dfrac{1}{2}}}}$
= ${\dfrac{2^{\dfrac{4}{3}}}{2^{\dfrac{1}{2}}}}$
= ${2^{\dfrac{4}{3}-\dfrac{1}{2}}}$
= ${2^{\dfrac{8-3}{6}}$
= ${2^{\dfrac{5}{6}}$
Here is a summary of the laws we follow while dividing expressions with exponents:
Last modified on October 26th, 2024