Previously, we have seen that we can rewrite square roots and cube roots with **rational exponents**. Since \left(a^\frac{1}{2}\right)^2 =a and \left(\sqrt{a}\right)^2=a, that means a^\frac{1}{2}=\sqrt{a}. A similar argument can be used for cube roots.

Consider the following statements:\begin{aligned} 81^\frac{1}{4} \cdot 81^\frac{1}{4} \cdot 81^\frac{1}{4} \cdot 81^\frac{1}{4} = \left(81^{\text{â¬š}}\right)^4&=81^{\text{â¬š}}=3^{\text{â¬š}}\\ \left(\sqrt[4]{81}\right)^4&=81=\left(81^{\text{â¬š}}\right)^4\\ \left(\sqrt[4]{81}\right)^3\cdot 81^\frac{1}{4}=81^{\text{â¬š}} \cdot 81^\frac{1}{4}&=81 \end{aligned}

- Fill in the blanks: 81^\frac{1}{â¬š}=3=\sqrt[â¬š]{81}
- What values for the missing exponents would make each statement true?
- What is the relationship between the values from each statement?
- Explain the meaning of 81^\frac{3}{4}.

Using the properties of exponents, we can express a^\frac{1}{n}, which represents one of n equal factors whose product equals a, multiplied by itself m times, using rational exponents as:

\displaystyle a^\frac{m}{n}

\bm{a}

is the base

\bm{\frac{m}{n}}

is the exponent

where, m and n are integers, and n \neq 0.

In general:

\displaystyle \left(\sqrt[n]{a}\right)^m=a^\frac{m}{n}=\sqrt[n]{a^m}

\bm{a}

is the base or radicand

\bm{m}

is the numerator or power

\bm{n}

is the denominator or index

We can use these rules for rewriting radicals along with the properties of exponents to simplify expressions involving radicals and rational exponents. Recall that the properties of exponents can be applied to integer exponents and rational exponents.

\text{Product of powers} | a^{m} \cdot a^{n} = a^{m+n} |

\text{Quotient of powers} | \dfrac{{a}^{m}}{{a}^{n}}=a^{m-n} |

\text{Power of a power} | \left(a^{m}\right)^{n} = a^{mn} |

\text{Power of a product} | \left(a b\right)^{m} = a^{m} \cdot b^{m} |

\text{Power of a quotient} | \left(\dfrac{a} {b}\right)^{m} =\dfrac {a^{m}} {b^{m}} |

\text{Identity exponent} | a^1=a |

\text{Zero exponent} | a^0=1 |

\text{Negative exponent} | a^{- {m}}=\dfrac{1}{a^{m}} |

Use the properties of exponents to define a rational exponent that would make the statement true: \left(\sqrt[3]{x^7}\right)^3=\left(x^{\frac{â¬š}{â¬š}}\right)^3=x^7

Worked Solution

Write each expression in an equivalent form using rational exponents. Assume all variables are positive.

a

\sqrt[5]{x^7}

Worked Solution

b

\left(\sqrt[4]{x^3y^5}\right)^{12}

Worked Solution

Write the following expressions in simplified radical form.

a

\left(bc\right)^{\frac{1}{5}}

Worked Solution

b

\left(\dfrac{81x}{625y^8}\right)^{\frac{1}{4}} assume all variables are positive.

Worked Solution

c

\left(-27a^{15}b^{27}\right)^\frac{1}{3}

Worked Solution

d

(-12x^5y^3)^{-\frac{1}{2}} assume all variables are positive.

Worked Solution

Idea summary

We can write radicals using rational exponents, for integer values of m and n, where n\neq 0:

\displaystyle \left(\sqrt[n]{a}\right)^m=a^\frac{m}{n}=\sqrt[n]{a^m}

\bm{a}

is the base or radicand

\bm{m}

is the numerator or power

\bm{n}

is the denominator or index

The properties of integer exponents can also be applied to rational exponents:

\text{Product of powers} | a^\frac{m}{n} \cdot a^\frac{p}{n} = a^\frac{m+p}{n} |

\text{Quotient of powers} | \dfrac{{a}^\frac{m}{n}}{{a}^\frac{p}{n}}=a^\frac{m-p}{n} |

\text{Power of a power} | (a^\frac{m}{n})^\frac{p}{q} = a^\frac{mp}{nq} |

\text{Power of a product} | (a b)^\frac{m}{n} = a^\frac{m}{n} \cdot b^\frac{m}{n} |

\text{Power of a quotient} | \left(\dfrac{a}{b}\right)^{\frac{m}{n}}=\dfrac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}} |

\text{Identity exponent} | a^1=a |

\text{Zero exponent} | a^0=1 |

\text{Negative exponent} | a^{-\frac{m}{n}}=\dfrac{1}{a^\frac{m}{n}} |