Q&A for How to Solve Decimal Exponents

You can do either one first, but it's easier to take the root first because it will give you a smaller number, which will be easier to work with in your head.

First, take the decimal and convert it to a fraction. The numerator of the fraction is an exponent and the denominator is going to be a root. For example, if you have a number like 8^2/3, you would take the cube root of 8 first, which is 2, then square that to get 4.

First convert 1.4 to 14/10, and reduce to 7/5. An exponent of 7/5 means first raising to the seventh power and then finding the fifth root (or first finding the fifth root and then raising to the seventh power). Thus, in this case you would raise 15 to the seventh power and then find that number's fifth root.

(.4)^4 = (.4)(.4)(.4)(.4) = .0256. Another way to do it is to express .4 as 4 x 10^-1. Thus (4 x 10^-1)^4 = 4^4 x 10^-4 = 256^-4 = .0256.

That's the seventh root of 2, and unless your mind is exceptionally good at doing calculations, it's not realistic to expect to solve that without a rather powerful calculator.

.3^1/3 means the cube root of (.3). 3^2/3 means the cube root of 3². Thus, with a fractional exponent, the numerator of the exponent indicates an actual exponent, and the denominator of the exponent indicates a root.

This is a unique situation. While the cube root of -8 does exist, the sixth root of -8 does not exist, because it's an imaginary number (√-2). So in this case the exponent 2/6 is not equivalent to the exponent 1/3. In a classroom setting an instructor can decide whether or not to define two such exponents as equivalent and then simplify.

0.174 = 174/1000 = 87/500. An exponent of 87/500 indicates the 500th root of the 87th power of whatever the base number is.

It would take a very powerful computer to solve that. First the exponent would have to be converted to the improper fraction 263/100. Then .833 would have to be expanded to the 263rd power, and then you'd have to find the 100th root of that number.

If the decimal is negative, like in 5^-2, simply ignore the negative sign and solve. Then divide the answer into 1. So, the above problem would equal 1/25. 4^-3 would equal 1/64. Fraction exponents are more complex. The numerator would be solved first, and the denominator is a root. So, 4^2/3 would be the cube root of 16.

An exponent of 1.4 is an exponent of 14/10, reduced to 7/5. That means raising a number to its seventh power, and then finding that number's fifth root. The seventh power of 2 is 128. The fifth root of 128 is 2.639.

What they're doing is breaking the exponent 3/4 into two parts, ¼ and 3, which when multiplied together make 3/4. So they're saying the base (81) will first be raised to the power of ¼ and then be raised again to the power of 3. In this case that means first finding the fourth root of 81, which is 3, and then raising that 3 to the power of 3, which is 27.

Multiply the number by itself one time fewer than the number in the exponent. For example, 8^6 means multiplying 8 times 8 five times. Put another way, 8 would be a factor six times in the answer: 8 x 8 x 8 x 8 x 8 x 8 = 262,144.

The fourth root is the square root of the square root.

2^(1/24) equals the 24th root of 2. You can't "simplify" it beyond that.

Raising x to the tenth power means using x as a factor ten times (which means multiplying x by itself nine -- not ten -- times). (1.047)^10 = 1.583.

3^(1/2) is the same thing as the square root of 3. √3 = 1.73.