Calculating a Decimal Exponent
Convert the decimal to a fraction. To convert a decimal to a fraction, consider place value. The denominator of the fraction will be the place value. The digits of the decimal will equal the numerator. For example, for the exponential expression 81 0.75 {\displaystyle 81^{0.75}} 81^{{0.75}}, you need to convert 0.75 {\displaystyle 0.75} 0.75 to a fraction. Since the decimal goes to the hundredths place, the corresponding fraction is 75 100 {\displaystyle {\frac {75}{100}}} {\frac {75}{100}}.
Simplify the fraction, if possible. Since you will be taking a root corresponding to the denominator of the exponent’s fraction, you want the denominator to be as small as possible. Do this by simplifying the fraction. If your fraction is a mixed number (that is, if your exponent was a decimal greater than 1), rewrite it as an improper fraction. For example, the fraction 75 100 {\displaystyle {\frac {75}{100}}} {\frac {75}{100}} reduces to 3 4 {\displaystyle {\frac {3}{4}}} {\frac {3}{4}}, So, 81 0.75 = 81 3 4 {\displaystyle 81^{0.75}=81^{\frac {3}{4}}} 81^{{0.75}}=81^{{{\frac {3}{4}}}}
Rewrite the exponent as a multiplication expression. To do this, turn the numerator into a whole number, and multiply it by the unit fraction. The unit fraction is the fraction with the same denominator, but with 1 as the numerator. For example, since 3 4 = 1 4 × 3 {\displaystyle {\frac {3}{4}}={\frac {1}{4}}\times 3} {\frac {3}{4}}={\frac {1}{4}}\times 3, you can rewrite the exponential expression as 81 1 4 × 3 {\displaystyle 81^{{\frac {1}{4}}\times 3}} 81^{{{\frac {1}{4}}\times 3}}.
Rewrite the exponent as a power of a power. Remember that multiplying two exponents is like taking the power of a power. So x 1 b × a {\displaystyle x^{\frac {1}{b}}\times a} x^{{{\frac {1}{b}}}}\times a becomes ( x 1 b ) a {\displaystyle (x^{\frac {1}{b}})^{a}} (x^{{{\frac {1}{b}}}})^{{a}}. For example, 81 1 4 × 3 = ( 81 1 4 ) 3 {\displaystyle 81^{{\frac {1}{4}}\times 3}=(81^{\frac {1}{4}})^{3}} 81^{{{\frac {1}{4}}\times 3}}=(81^{{{\frac {1}{4}}}})^{{3}}.
Rewrite the base as a radical expression. Taking a number by a rational exponent is equal to taking the appropriate root of the number. So, rewrite the base and its first exponent as a radical expression. For example, since 81 1 4 = 81 4 {\displaystyle 81^{\frac {1}{4}}={\sqrt[{4}]{81}}} 81^{{{\frac {1}{4}}}}={\sqrt[ {4}]{81}}, you can rewrite the expression as ( 81 4 ) 3 {\displaystyle ({\sqrt[{4}]{81}})^{3}} ({\sqrt[ {4}]{81}})^{{3}}.
Calculate the radical expression. Remember that the index (the small number outside the radical sign) tells you which root you are looking for. If the numbers are cumbersome, the best way to do this is using the y x {\displaystyle {\sqrt[{x}]{y}}} {\sqrt[ {x}]{y}} feature on a scientific calculator. For example, to calculate 81 4 {\displaystyle {\sqrt[{4}]{81}}} {\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} 3\times 3\times 3\times 3=81, you know that 81 4 = 3 {\displaystyle {\sqrt[{4}]{81}}=3} {\sqrt[ {4}]{81}}=3. So, the exponential expression now becomes 3 3 {\displaystyle 3^{3}} 3^{{3}}.
Calculate the remaining exponent. You should now have a whole number as an exponent, so calculating should be straightforward. You can always use a calculator if the numbers are too large. For example, 3 3 = 3 × 3 × 3 = 27 {\displaystyle 3^{3}=3\times 3\times 3=27} 3^{{3}}=3\times 3\times 3=27. So, 81 0.75 = 27 {\displaystyle 81^{0.75}=27} 81^{{0.75}}=27.
Solving a Sample Problem
Calculate the following exponential expression: 256 2.25 {\displaystyle 256^{2.25}} 256^{{2.25}}.
Convert the decimal to a fraction. Since 2.25 {\displaystyle 2.25} 2.25 is greater than 1, the fraction will be a mixed number. The decimal 0.25 {\displaystyle 0.25} 0.25 is equal to 25 100 {\displaystyle {\frac {25}{100}}} {\frac {25}{100}}, so 2.25 = 2 25 100 {\displaystyle 2.25=2{\frac {25}{100}}} 2.25=2{\frac {25}{100}}.
Simplify the fraction, if possible. You should also convert any mixed numbers to improper fractions. Since 25 100 {\displaystyle {\frac {25}{100}}} {\frac {25}{100}} reduces to 1 4 {\displaystyle {\frac {1}{4}}} {\frac {1}{4}}, 2 25 100 = 2 1 4 {\displaystyle 2{\frac {25}{100}}=2{\frac {1}{4}}} 2{\frac {25}{100}}=2{\frac {1}{4}}. Converting to an improper fraction, you have 9 4 {\displaystyle {\frac {9}{4}}} {\frac {9}{4}}. So, 256 2.25 = 256 9 4 {\displaystyle 256^{2.25}=256^{\frac {9}{4}}} 256^{{2.25}}=256^{{{\frac {9}{4}}}}.
Rewrite the exponent as a multiplication expression. Since 9 4 = 1 4 × 9 {\displaystyle {\frac {9}{4}}={\frac {1}{4}}\times 9} {\frac {9}{4}}={\frac {1}{4}}\times 9, you can rewrite the expression as 256 1 4 × 9 {\displaystyle 256^{{\frac {1}{4}}\times 9}} 256^{{{\frac {1}{4}}\times 9}}.
Rewrite the exponent as a power of a power. So, 256 1 4 × 9 = ( 256 1 4 ) 9 {\displaystyle 256^{{\frac {1}{4}}\times 9}=(256^{\frac {1}{4}})^{9}} 256^{{{\frac {1}{4}}\times 9}}=(256^{{{\frac {1}{4}}}})^{{9}}.
Rewrite the base as a radical expression. 256 1 4 = 256 4 {\displaystyle 256^{\frac {1}{4}}={\sqrt[{4}]{256}}} 256^{{{\frac {1}{4}}}}={\sqrt[ {4}]{256}}, so you can rewrite the expression as ( 256 4 ) 9 {\displaystyle ({\sqrt[{4}]{256}})^{9}} ({\sqrt[ {4}]{256}})^{{9}}.
Calculate the radical expression. 256 4 = 4 {\displaystyle {\sqrt[{4}]{256}}=4} {\sqrt[ {4}]{256}}=4. So, the expression is now ( 4 ) 9 {\displaystyle (4)^{9}} (4)^{{9}}.
Calculate the remaining exponent. ( 4 ) 9 = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 262 , 144 {\displaystyle (4)^{9}=4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4=262,144} (4)^{{9}}=4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4=262,144. So, 256 2.25 = 262 , 144 {\displaystyle 256^{2.25}=262,144} 256^{{2.25}}=262,144
Understanding Exponents
Recognize an exponential expression. An exponential expression has a base and an exponent. The base is the large number in the expression. The exponent is the smaller number. For example, in the expression 3 4 {\displaystyle 3^{4}} 3^{{4}}, 3 {\displaystyle 3} 3 is the base and 4 {\displaystyle 4} 4 is the exponent.
Identify the parts of an exponential expression. The base is the number that is being multiplied. The exponent tells you how many times the base is being used as a factor in the expression. For example, 3 4 = 3 × 3 × 3 × 3 = 81 {\displaystyle 3^{4}=3\times 3\times 3\times 3=81} 3^{{4}}=3\times 3\times 3\times 3=81.
Identify a rational exponent. A rational exponent is also called a fractional exponent. It is an exponent that takes the form of a fraction. For example, 4 1 2 {\displaystyle 4^{\frac {1}{2}}} 4^{{{\frac {1}{2}}}}.
Understand the relationship between radicals and rational exponents. Taking a number to the 1 2 {\displaystyle {\frac {1}{2}}} {\frac {1}{2}} power is like taking the square root of the number. So, x 1 2 = x {\displaystyle x^{\frac {1}{2}}={\sqrt {x}}} x^{{{\frac {1}{2}}}}={\sqrt {x}}. The same is true for other roots and exponents. The denominator of the exponent will tell you which root to take: x 1 3 = x 3 {\displaystyle x^{\frac {1}{3}}={\sqrt[{3}]{x}}} x^{{{\frac {1}{3}}}}={\sqrt[ {3}]{x}} x 1 4 = x 4 {\displaystyle x^{\frac {1}{4}}={\sqrt[{4}]{x}}} x^{{{\frac {1}{4}}}}={\sqrt[ {4}]{x}} x 1 5 = x 5 {\displaystyle x^{\frac {1}{5}}={\sqrt[{5}]{x}}} x^{{{\frac {1}{5}}}}={\sqrt[ {5}]{x}} For example, 81 1 4 = 81 4 = 3 {\displaystyle 81^{\frac {1}{4}}={\sqrt[{4}]{81}}=3} 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} 3\times 3\times 3\times 3=81
Understand the exponential law of powers of powers. This law says that ( x a ) b = x a b {\displaystyle (x^{a})^{b}=x^{ab}} (x^{{a}})^{{b}}=x^{{ab}}. In other words taking an exponent to another power is the same as multiplying the two exponents. When working with rational exponents, this law looks like x a b = ( x 1 b ) a {\displaystyle x^{\frac {a}{b}}=(x^{\frac {1}{b}})^{a}} x^{{{\frac {a}{b}}}}=(x^{{{\frac {1}{b}}}})^{{a}}, since 1 b × a = a b {\displaystyle {\frac {1}{b}}\times a={\frac {a}{b}}} {\frac {1}{b}}\times a={\frac {a}{b}}. 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.
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