4. Radical Functions & Rational Exponents

Lesson

Let's review the laws of exponents. It's important to remember the order of operations when we're simplifying these expressions.

Laws of exponents

- The product of powers property: $a^m\times a^n=a^{m+n}$
`a``m`×`a``n`=`a``m`+`n` - The quotient of powers property: $a^m\div a^n=a^{m-n}$
`a``m`÷`a``n`=`a``m`−`n` - The zero exponent property: $a^0=1$
`a`0=1 - The power of a power property: $\left(a^m\right)^n=a^{mn}$(
`a``m`)`n`=`a``m``n` - The negative exponent definition: $a^{-m}=\frac{1}{a^m}$
`a`−`m`=1`a``m`

A question may have any combination of laws of exponents. We just need to simplify it step by step, making sure we follow the order of operations.

**Simplify:** $p^7\div p^3\times p^5$`p`7÷`p`3×`p`5

**Think**: We need to apply the exponent division and exponent multiplication laws.

**Do**:

$p^7\div p^3\times p^5$p7÷p3×p5 |
$=$= | $p^{7-3+5}$p7−3+5 |

$=$= | $p^9$p9 |

**Reflect:** We can choose to do this in more steps by first doing $p^{7-3}\times p^5=p^4\times p^5$`p`7−3×`p`5=`p`4×`p`5 and then getting our final answer of $p^9$`p`9.

Question 2

Simplify: $\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(`u``x`+3)3`u``x`+1

**Think:** We need to simplify the numerator using the power of a power property, then apply the quotient property.

**Do:**

$\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(ux+3)3ux+1 |
$=$= | $\frac{u^{3\left(x+3\right)}}{u^{x+1}}$u3(x+3)ux+1 |
Simplify the numerator using the power of a power property |

$=$= | $\frac{u^{3x+9}}{u^{x+1}}$u3x+9ux+1 |
Apply the distributive property | |

$=$= | $u^{3x+9-\left(x+1\right)}$u3x+9−(x+1) |
Use the quotient property and subtract the powers | |

$=$= | $u^{3x+9-x-1}$u3x+9−x−1 |
Simplify by collecting the like terms | |

$=$= | $u^{2x+8}$u2x+8 |

Express $\left(4^p\right)^4$(4`p`)4 with a prime number base in exponential form.

**Think:** We could express $4$4 as $2^2$22 which has a prime number base.

**Do:**

$\left(4^p\right)^4$(4p)4 |
$=$= | $4^{4p}$44p |
Use the power of a power property |

$=$= | $\left(2^2\right)^{4p}$(22)4p |
Use the fact that $4=2^2$4=22 | |

$=$= | $2^{8p}$28p |
Use the power of a power property |

**Reflect:** This skill will become increasingly important as we look at simplifying expressions with related bases such as $2^{3p}\times\left(4^p\right)^4$23`p`×(4`p`)4.

Simplify $20m^6\div5m^{13}\times9m^2$20`m`6÷5`m`13×9`m`2, expressing your answer in positive exponential form.

**Think: **Let's express this as a fraction so the powers are on the numerator and the denominator for easy comparison.

**Do:**

$\frac{20m^6}{5m^{13}}\times9m^2$20m65m13×9m2 |
$=$= | $\frac{4}{m^7}\times9m^2$4m7×9m2 |
Simplify the fraction using the quotient property |

$=$= | $\frac{36m^2}{m^7}$36m2m7 |
Simplify the multiplication | |

$=$= | $36m^{-5}$36m−5 |
Use the quotient property - this step is sometimes omitted | |

$=$= | $\frac{36}{m^5}$36m5 |
Write as a positive exponent |

Simplify $\frac{\left(x^2\right)^6}{\left(x^2\right)^2}$(`x`2)6(`x`2)2

Simplify $\left(u^9\times u^5\div u^{19}\right)^2$(`u`9×`u`5÷`u`19)2, expressing your answer in positive exponential form.

Express $\left(5y^3\right)^{-3}$(5`y`3)−3 with a positive exponent.