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How to Convert Square Roots to Exponents

Finding square roots and converting them to exponents is a relatively common operation in algebra. Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical.

When you find square roots, the symbol for that operation is a radical, which looks like this:

image0.png

When changing from radical form to fractional exponents, remember these basic forms:

  • The nth root of a can be written as a fractional exponent with a raised to the reciprocal of that power.

    image1.png
  • When the nth root of

    image2.png
  • is taken, it’s raised to the 1/n power. When a power is raised to another power, you multiply the powers together, and so the m (otherwise written as m/1) and the 1/n are multiplied together.

    image3.png

Use fractions in the powers to indicate that the expression stands for a root or a radical.

Here are some examples of changing radical forms to fractional exponents:

image4.png

When raising a power to a power, you multiply the exponents, but the bases have to be the same.

Because raising a power to a power means that you multiply exponents (as long as the bases are the same), you can simplify the following expressions:

image5.png

Leave the exponent as 9/4. Don’t write it as a mixed number.

The following example can’t be combined because the bases are not the same:

image6.png
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