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Radicals and rational exponents | Lesson

A guide to radicals and rational exponents on the digital SAT

What are radicals and rational exponents?

Exponential expressions are algebraic expressions with a coefficient, one or more variables, and one or more exponents. For example, in the expression start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript:
  • start color #7854ab, 3, end color #7854ab is the coefficient.
  • start color #ca337c, x, end color #ca337c is the base.
  • start color #208170, 4, end color #208170 is the exponent.
In start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript, start color #7854ab, 3, end color #7854ab is multiplied by start color #ca337c, x, end color #ca337c start color #208170, 4, end color #208170 times:
start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript, equals, start color #7854ab, 3, dot, end color #7854ab, start color #208170, left parenthesis, end color #208170, start color #ca337c, x, end color #ca337c, start color #208170, dot, end color #208170, start color #ca337c, x, end color #ca337c, start color #208170, dot, end color #208170, start color #ca337c, x, end color #ca337c, start color #208170, dot, end color #208170, start color #ca337c, x, end color #ca337c, start color #208170, right parenthesis, end color #208170
An expression can also be raised to an exponent. For example, for left parenthesis, start color #ca337c, 3, x, end color #ca337c, right parenthesis, start superscript, start color #208170, 4, end color #208170, end superscript, the expression start color #ca337c, 3, x, end color #ca337c is multiplied by itself start color #208170, 4, end color #208170 times:
left parenthesis, start color #ca337c, 3, x, end color #ca337c, right parenthesis, start superscript, start color #208170, 4, end color #208170, end superscript, equals, start color #ca337c, 3, x, end color #ca337c, start color #208170, dot, end color #208170, start color #ca337c, 3, x, end color #ca337c, start color #208170, dot, end color #208170, start color #ca337c, 3, x, end color #ca337c, start color #208170, dot, end color #208170, start color #ca337c, 3, x, end color #ca337c, equals, start color #7854ab, 81, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript
Notice how 3, x, start superscript, 4, end superscript, does not equal, left parenthesis, 3, x, right parenthesis, start superscript, 4, end superscript !
Rational exponents refer to exponents that are/can be represented as fractions: start fraction, 1, divided by, 2, end fraction, 3, and minus, start fraction, 2, divided by, 3, end fraction are all considered rational exponents. Radicals are another way to write rational exponents. For example, x, start superscript, start superscript, start fraction, 1, divided by, 2, end fraction, end superscript, end superscript and square root of, x, end square root are equivalent.
In this lesson, we'll:
  1. Review the rules of exponent operations with integer exponents
  2. Apply the rules of exponent operations to rational exponents
  3. Make connections between equivalent rational and radical expressions
You can learn anything. Let's do this!

What are the rules of exponent operations?

Powers of products & quotients (integer exponents)

Khan Academy video wrapper
Powers of products & quotients (integer exponents)See video transcript

The rules of exponent operations

Adding and subtracting exponential expressions

When adding and subtracting exponential expressions, we're essentially combining like terms. That means we can only combine exponential expressions with both the same base and the same exponent.
axn±bxn=(a±b)xn\begin{aligned} ax^n\pm bx^n &= (a\pm b )x^n \end{aligned}

Multiplying and dividing exponential expressions

When multiplying two exponential expressions with the same base, we keep the base the same, multiply the coefficients, and add the exponents. Similarly, when dividing two exponential expressions with the same base, we keep the base the same and subtract the exponents.
axmbxn=abxm+naxmbxn=abxmn\begin{aligned} ax^m \cdot bx^n &= ab\cdot x^{m+n} \\\\ \dfrac{ax^m}{bx^n} &= \dfrac{a}{b}\cdot x^{m-n} \end{aligned}
When multiplying or dividing exponential expressions with the same exponent but different bases, we multiply or divide the bases and keep the exponents the same.
xnyn=(xy)nxnyn=(xy)n\begin{aligned} x^n\cdot y^n &= (xy)^n \\\\ \dfrac{x^n}{y^n} &= \left(\dfrac{x}{y}\right)^n \end{aligned}

Raising an exponential expression to an exponent and change of base

When raising an exponential expression to an exponent, raise the coefficient of the expression to the exponent, keep the base the same, and multiply the two exponents.
left parenthesis, a, x, start superscript, m, end superscript, right parenthesis, start superscript, n, end superscript, equals, a, start superscript, n, end superscript, dot, x, start superscript, m, n, end superscript
When the bases are numbers, we can use a similar rule to change the base of an exponential expression.
left parenthesis, a, start superscript, b, end superscript, right parenthesis, start superscript, n, end superscript, equals, a, start superscript, b, n, end superscript
This is useful for questions with multiple terms that need to be written in the same base.

Negative exponents

A base raised to a negative exponent is equivalent to 1 divided by the base raised to the
opposite
of the exponent.
x, start superscript, minus, n, end superscript, equals, start fraction, 1, divided by, x, start superscript, n, end superscript, end fraction

Zero exponent

A nonzero base raised to an exponent of 0 is equal to 1.
x, start superscript, 0, end superscript, equals, 1, comma, x, does not equal, 0

How do the rules of exponent operations apply to rational exponents?

Every rule that applies to integer exponents also applies to rational exponents.

Try it!

try: divide two rational expressions
In order to divide 12, x, start superscript, start superscript, start fraction, 5, divided by, 2, end fraction, end superscript, end superscript by 3, x, start superscript, start superscript, start fraction, 1, divided by, 2, end fraction, end superscript, end superscript, we
the coefficients and
the exponents of x.
start fraction, 12, x, start superscript, start superscript, start fraction, 5, divided by, 2, end fraction, end superscript, end superscript, divided by, 3, x, start superscript, start superscript, start fraction, 1, divided by, 2, end fraction, end superscript, end superscript, end fraction, equals


Try: raise an exponential expression to an exponent
To calculate left parenthesis, 2, y, start superscript, start superscript, start fraction, 4, divided by, 3, end fraction, end superscript, end superscript, right parenthesis, cubed, we
and
the exponents start fraction, 4, divided by, 3, end fraction and 3.
left parenthesis, 2, y, start superscript, start superscript, start fraction, 4, divided by, 3, end fraction, end superscript, end superscript, right parenthesis, cubed, equals


How are radicals and fractional exponents related?

Rewriting roots as rational exponents

Khan Academy video wrapper
Rewriting roots as rational exponentsSee video transcript

Roots and rational exponents

Squares and square roots are inverse operations: they "undo" each other. For example, if we take the square root of 3 squared, we get square root of, 3, squared, end square root, equals, 3.
The reason for this becomes more apparent when we rewrite square root as a fractional exponent: square root of, x, end square root, equals, x, start superscript, start superscript, start fraction, 1, divided by, 2, end fraction, end superscript, end superscript, and square root of, 3, squared, end square root, equals, left parenthesis, 3, squared, right parenthesis, start superscript, start superscript, start fraction, 1, divided by, 2, end fraction, end superscript, end superscript, equals, 3, start superscript, 1, end superscript.
When rewriting a radical expression as a fractional exponent, any exponent under the radical symbol (square root of, empty space, end square root) becomes the numerator of the fractional exponent, and the value to the left of the radical symbol (e.g., root, start index, start color #7854ab, 3, end color #7854ab, end index) becomes the denominator of the fractional exponent. Square root is equivalent to root, start index, 2, end index.
root, start index, n, end index, equals, x, start superscript, start superscript, start fraction, m, divided by, n, end fraction, end superscript, end superscript
All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents. Similarly, for radical expressions:
xnyn=xynxnyn=xyn\begin{aligned} \sqrt[n]{x} \cdot \sqrt[n]{y} &= \sqrt[n]{xy} \\\\ \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} &= \sqrt[n]{\dfrac{x}{y}} \end{aligned}
When working with radical expressions with the same radical, we can choose whether to convert to fractional exponents first or multiply what's under the radical symbol first to our advantage.

Try it!

Try: determine equivalent expressions
Determine whether each of the radical expressions below is equivalent to x, start superscript, start superscript, start fraction, 3, divided by, 2, end fraction, end superscript, end superscript, y, start superscript, start superscript, start fraction, 1, divided by, 3, end fraction, end superscript, end superscript.
 
Equivalent
Not equivalent
square root of, x, cubed, end square root, dot, cube root of, y, end cube root
square root of, x, y, end square root
square root of, x, cubed, y, end square root
root, start index, 6, end index


Your turn!

Practice: multiply rational expressions
Which of the following is equivalent to 2, x, cubed, dot, 3, x, start superscript, 5, end superscript ?
Choose 1 answer:
Choose 1 answer:


Practice: change bases
If a, start superscript, start superscript, start fraction, b, divided by, 2, end fraction, end superscript, end superscript, equals, 25 for positive integers a and b, what is one possible value of b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Practice: raise to a negative exponent
If n, start superscript, start superscript, minus, start fraction, 1, divided by, 3, end fraction, end superscript, end superscript, equals, x, where n, is greater than, 0, what is n in terms of x ?
Choose 1 answer:
Choose 1 answer:


Practice: simplify radical expressions
start fraction, cube root of, 8, x, start superscript, 8, end superscript, y, start superscript, 6, end superscript, end cube root, divided by, square root of, 4, x, squared, y, start superscript, 6, end superscript, end square root, end fraction
Which of the following is equivalent to the expression above?
Choose 1 answer:
Choose 1 answer:


Things to remember

Adding and subtracting exponential expressions:
axn±bxn=(a±b)xn\begin{aligned} ax^n\pm bx^n &= (a\pm b )x^n \end{aligned}
Multiplying and dividing exponential expressions:
axmbxn=abxm+naxmbxn=abxmnxnyn=(xy)nxnyn=(xy)n\begin{aligned} ax^m \cdot bx^n &= ab\cdot x^{m+n} \\\\ \dfrac{ax^m}{bx^n} &= \dfrac{a}{b}\cdot x^{m-n} \\\\ x^n\cdot y^n &= (xy)^n \\\\ \dfrac{x^n}{y^n} &= \left(\dfrac{x}{y}\right)^n \end{aligned}
Raising an exponential expression to an exponent and change of base:
(axm)n=anxmn(ab)n=abn\begin{aligned} \left(ax^m\right)^n &= a^n \cdot x^{mn} \\\\ (a^b)^n &=a^{bn} \end{aligned}
Negative exponent:
x, start superscript, minus, n, end superscript, equals, start fraction, 1, divided by, x, start superscript, n, end superscript, end fraction
Zero exponent:
x, start superscript, 0, end superscript, equals, 1, comma, x, does not equal, 0
All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents.
xmn=xmnxnyn=xynxnyn=xyn\begin{aligned} \sqrt[n]{x^m} &= x^{^{\scriptsize \dfrac{m}{n}}} \\\\ \sqrt[n]{x} \cdot \sqrt[n]{y} &= \sqrt[n]{xy} \\\\ \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} &= \sqrt[n]{\dfrac{x}{y}} \end{aligned}

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