Previously, we also looked at the property below which can be used to simplify rational expressions.
$\sqrt{ab}=\sqrt{a}\times\sqrt{b}$√ab=√a×√b and $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$√a×√b=√ab
where $a\ge0$a≥0 and $b\ge0$b≥0
The key idea is that if we can break the radicand (argument) into two factors where one is a perfect square, then we can use the fact that $\sqrt{a^2}=a$√a2=a to simplify. This can be extended to higher powers and radicands with variables as well.
The trick of course is to recognize these perfect squares or cubes whenever they occur.
Here is a table showing the first $12$12 squares:
$2^2$22 | $3^2$32 | $4^2$42 | $5^2$52 | $6^2$62 | $7^2$72 | $8^2$82 | $9^2$92 | $10^2$102 | $11^2$112 | $12^2$122 |
---|---|---|---|---|---|---|---|---|---|---|
$4$4 | $9$9 | $16$16 | $25$25 | $36$36 | $49$49 | $64$64 | $81$81 | $100$100 | $121$121 | $144$144 |
Here is a table showing the first $6$6 cubes:
$1^3$13 | $2^3$23 | $3^3$33 | $4^3$43 | $5^3$53 | $6^3$63 |
---|---|---|---|---|---|
$1$1 | $8$8 | $27$27 | $64$64 | $125$125 | $216$216 |
Simplify $\sqrt{8}$√8.
Think: If we can find any factors of $8$8 that are perfect squares, then we can simplify the expression using the fact that $\sqrt{a^2b}=a\sqrt{b}$√a2b=a√b.
Do: The factors of $8$8 are $1$1, $2$2, $4$4, and $8$8. Let's use the perfect square $4$4 to rewrite the expression and simplify.
$\sqrt{8}$√8 | $=$= | $\sqrt{4\times2}$√4×2 | (Replace $8$8 with two factors) |
$=$= | $\sqrt{4}\times\sqrt{2}$√4×√2 | (Use the fact that $\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$√a×b=√a×√b) | |
$=$= | $\sqrt{2^2}\times\sqrt{2}$√22×√2 | (Rewrite $4$4 as $2^2$22) | |
$=$= | $2\sqrt{2}$2√2 | (Use the fact that $\sqrt{a^2}=a$√a2=a) |
Simplify $\sqrt[3]{72}$^{3}√72.
Think: If we can find any factors of $72$72 that are perfect cubes, then we can simplify the expression using the fact that $\sqrt[3]{a^3}=a$^{3}√a3=a.
Do: The factors of $72$72 are $1$1, $2$2, $3$3, $4$4, $6$6, $8$8, $9$9, $12$12, $18$18, $24$24, $36$36, and $72$72. Let's use the perfect cube $8$8 to rewrite the expression and simplify.
$\sqrt[3]{72}$^{3}√72 | $=$= | $\sqrt[3]{8\times9}$^{3}√8×9 | (Replace $72$72 with two factors) |
$=$= | $\sqrt[3]{8}\times\sqrt[3]{9}$^{3}√8×^{3}√9 | (Use the fact that $\sqrt[3]{a\times b}=\sqrt[3]{a}\times\sqrt[3]{b}$^{3}√a×b=^{3}√a×^{3}√b) | |
$=$= | $\sqrt[3]{2^3}\times\sqrt[3]{9}$^{3}√23×^{3}√9 | (Rewrite $8$8 as $2^3$23) | |
$=$= | $2\sqrt[3]{9}$2^{3}√9 | (Use the fact that $\sqrt[3]{a^3}=a$^{3}√a3=a) |
Simplify $\sqrt{150}$√150.
We have seen that we can convert between a radical expression and an expression with rational exponents. We can use this property to see how we can simplify expressions involving radicals.
Your laws of exponents may be helpful when simplifying algebraic roots. Here are a couple of handy ones:
Simplify: $\sqrt[3]{a^3b^6}$^{3}√a3b6.
Think: We can use the fractional exponent, power of a product, and power of a power rules to simplify this expression.
Do:
$\sqrt[3]{a^3}$^{3}√a3 | $=$= | $\left(a^3b^6\right)^{\frac{1}{3}}$(a3b6)13 | Using the fractional exponent property |
$=$= | $a^{3\times\frac{1}{3}}b^{6\times\frac{1}{3}}$a3×13b6×13 | Using the power of a power and power of a product properties | |
$=$= | $a^1b^2$a1b2 | Evaluate the multiplication | |
$=$= | $ab^2$ab2 | Simplify |
Simplify: $\sqrt{50b^8}$√50b8.
Think: First we can break the product into the numerical and algebraic component, then we can use the fractional exponent, power of a product, and power of a power rules to simplify this expression.
Do:
$\sqrt{50b^9}$√50b9 | $=$= | $\sqrt{50}\sqrt{b^9}$√50√b9 | Use $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$√ab=√a×√b |
$=$= | $\sqrt{25}\sqrt{2}\sqrt{b^8}\sqrt{b}$√25√2√b8√b | Use $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$√ab=√a×√b to break out perfect squares | |
$=$= | $5\sqrt{2}b^4\sqrt{b}$5√2b4√b | Use the fact that $\sqrt{a^2}=a$√a2=a | |
$=$= | $5b^4\sqrt{2b}$5b4√2b | Use $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$√a×√b=√ab |
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.
Assuming that $x$x and $y$y both positive, simplify the expression $\sqrt{3^2x^{14}y^{20}}$√32x14y20.
Simplify $\sqrt[3]{b^3x^9}$^{3}√b3x9.
Assuming that $x$x represents a positive number, simplify the expression $\sqrt{\frac{49x^4}{64}}$√49x464:
Rewrite expressions involving radicals and rational exponents using the properties of exponents.