How to Solve Exponents With Decimals

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Exponents can come in one of several forms, such as a whole number, fraction or decimal. A whole number is a number without a fraction or decimal. A decimal contains a portion of a number to the right of the decimal point. A fractional exponent contains a numerator and a denominator. The numerator is the power number of which to raise the base, which is the number with the exponent. The denominator is the root number to take of the base. Exponents with decimals can be converted into fractional exponents and solved in a series of steps to make it easier to solve an expression.

  • Determine an expression that contains a decimal exponent. For the following example, use 9^1.5.

  • Separate the decimal exponent into a whole number and a decimal. In the example, this results in 1 and 0.5.

  • Rewrite the expression as a product of two terms -- one with the base raised to an exponent that contains the whole number and one with the base raised to an exponent that contains the decimal. In the example, this results in the product of the two terms 9^1 x 9^0.5.

  • Convert the decimal exponent into a fraction by placing the number to the right of the decimal as a numerator over a denominator that corresponds to the number of places the number extends to the right of the decimal. In the example, the decimal exponent extends one place to the right of the decimal, which is the tenths place, so place 5 as a numerator over 10 as a denominator. This results in an exponent of 5/10, which leaves the expression 9^1 x 9^(5/10).

  • Divide both the numerator and denominator of the fractional exponent by the largest number that divides evenly into both to reduce the fractional exponent to its lowest terms, if possible. In the example, the number 5 is the largest number that divides evenly into 5 and 10, so divide 5 by 5, which equals 1, and divide 10 by 5, which equals 2. This results in a fractional exponent of 1/2, which leaves the expression 9^1 x 9^(1/2).

  • Calculate the term of the expression with the whole-number exponent. In the example, calculate 9^1, which equals 9. This leaves 9 x 9^(1/2).

  • Calculate the term of the expression with the fractional exponent. Take the root of the base equal to the number in the denominator. In the example, the denominator is 2, so take the square root of 9. This equals 3, which leaves 9 x 3^1.

  • Raise the result to the power of the numerator that remains in the fractional exponent. In the example, 1 remains as the numerator in the fractional exponent, so raise 3 to the power of 1, which equals 3. This leaves the expression 9 x 3.

  • Multiply the remaining terms in the expression. In the example, multiply 9 by 3, which equals 27.

References

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