Simplify expressions involving algebraic roots
We've already looked at evaluating square roots and cube roots in Know Your Numerical Roots. However, we can also evaluate roots of algebraic terms. The answers to algebraic roots are often just simplified algebraic terms, so don't be expecting nice neat whole number answers.
Your index laws may be helpful when evaluating algebraic roots. Here are a couple of handy ones:
- Fractional index law: $a^{\frac{x}{y}}=\sqrt[y]{a^x}$axy=y√ax
- Power of a power rule: $\left(a^x\right)^y=a^{xy}$(ax)y=axy
Examples
Question 1
Simplify: $\sqrt[3]{a^3}$3√a3.
Think: We can use the fractional index and power of of a power rules to simplify this expression.
Do:
| $\sqrt[3]{a^3}$3√a3 | $=$= | $\left(a^3\right)^{\frac{1}{3}}$(a3)13 | Using the fractional index rule |
| $=$= | $a^{3\times\frac{1}{3}}$a3×13 | Using the power of a power rule | |
| $=$= | $a^1$a1 | ||
| $=$= | $a$a |
Remember that when we raise a negative number to an even power, it becomes a positive number. For instance, $\left(-5\right)^2=25$(−5)2=25. This means that $\sqrt{(-5)^2}=\sqrt{25}=5$√(−5)2=√25=5.
If we now consider the algebraic expression $\sqrt{a^2}$√a2, the power of a power rule indicates that this should simplify to $a$a. As you can see above, however, this is not the case if $a$a is a negative number!
So be careful when simplifying even powers and roots of algebraic expressions - make sure to think about whether or not the variable could represent a negative number.
Question 2
Assuming that $x$x and $y$y both positive, simplify the expression $\sqrt{5^2x^{14}y^{20}}$√52x14y20.
Question 3
Simplify $\sqrt[3]{b^3x^9}$3√b3x9.
Question 4
Assuming that $x$x represents a positive number, simplify the expression $\sqrt{\frac{49x^4}{64}}$√49x464: