For example, for the exponential expression 810. 75{\displaystyle 81^{0. 75}}, you need to convert 0. 75{\displaystyle 0. 75} to a fraction. Since the decimal goes to the hundredths place, the corresponding fraction is 75100{\displaystyle {\frac {75}{100}}}.
For example, the fraction 75100{\displaystyle {\frac {75}{100}}} reduces to 34{\displaystyle {\frac {3}{4}}}, So, 810. 75=8134{\displaystyle 81^{0. 75}=81^{\frac {3}{4}}}
For example, since 34=14×3{\displaystyle {\frac {3}{4}}={\frac {1}{4}}\times 3}, you can rewrite the exponential expression as 8114×3{\displaystyle 81^{{\frac {1}{4}}\times 3}}.
For example, 8114×3=(8114)3{\displaystyle 81^{{\frac {1}{4}}\times 3}=(81^{\frac {1}{4}})^{3}}.
For example, since 8114=814{\displaystyle 81^{\frac {1}{4}}={\sqrt[{4}]{81}}}, you can rewrite the expression as (814)3{\displaystyle ({\sqrt[{4}]{81}})^{3}}. [3] X Research source
For example, to calculate 814{\displaystyle {\sqrt[{4}]{81}}}, you need to determine which number multiplied 4 times is equal to 81. Since 3×3×3×3=81{\displaystyle 3\times 3\times 3\times 3=81}, you know that 814=3{\displaystyle {\sqrt[{4}]{81}}=3}. So, the exponential expression now becomes 33{\displaystyle 3^{3}}.
For example, 33=3×3×3=27{\displaystyle 3^{3}=3\times 3\times 3=27}. So, 810. 75=27{\displaystyle 81^{0. 75}=27}.
The decimal 0. 25{\displaystyle 0. 25} is equal to 25100{\displaystyle {\frac {25}{100}}}, so 2. 25=225100{\displaystyle 2. 25=2{\frac {25}{100}}}.
Since 25100{\displaystyle {\frac {25}{100}}} reduces to 14{\displaystyle {\frac {1}{4}}}, 225100=214{\displaystyle 2{\frac {25}{100}}=2{\frac {1}{4}}}. Converting to an improper fraction, you have 94{\displaystyle {\frac {9}{4}}}. So, 2562. 25=25694{\displaystyle 256^{2. 25}=256^{\frac {9}{4}}}.
For example, in the expression 34{\displaystyle 3^{4}}, 3{\displaystyle 3} is the base and 4{\displaystyle 4} is the exponent.
For example, 34=3×3×3×3=81{\displaystyle 3^{4}=3\times 3\times 3\times 3=81}.
For example, 412{\displaystyle 4^{\frac {1}{2}}}.
x13=x3{\displaystyle x^{\frac {1}{3}}={\sqrt[{3}]{x}}} x14=x4{\displaystyle x^{\frac {1}{4}}={\sqrt[{4}]{x}}} x15=x5{\displaystyle x^{\frac {1}{5}}={\sqrt[{5}]{x}}} For example, 8114=814=3{\displaystyle 81^{\frac {1}{4}}={\sqrt[{4}]{81}}=3}. You know that 3 is the fourth root of 81 since 3×3×3×3=81{\displaystyle 3\times 3\times 3\times 3=81}
When working with rational exponents, this law looks like xab=(x1b)a{\displaystyle x^{\frac {a}{b}}=(x^{\frac {1}{b}})^{a}}, since 1b×a=ab{\displaystyle {\frac {1}{b}}\times a={\frac {a}{b}}}. [9] X Research source It doesn’t actually matter whether you do the root or the exponent part of the problem first. However, taking the root first will give you a smaller number to work with, which usually makes the problem easier to solve. [10] X Expert Source David JiaAcademic Tutor Expert Interview. 14 January 2021.
