Lesson Objectives
- Learn how to identify the parts of a radical
- Learn how to simplify radical expressions
- Learn how to work with rational exponents
Radical Expressions
In this lesson, we will learn about radical expressions and rational exponents. Most commonly, we will start out by learning about the square root operation. The square root of a number is a number that multiplies by itself to give the number. The square root operation is the inverse operation of the squaring operation. If we want to ask for the square root of a number, we can place the number underneath a square root symbol. For example, if we wanted to ask for the square root of the number 9, we could write this as: $$\sqrt{9}$$ Since 3 squared is 9, the square root of 9 is 3. $$3^2=9$$ $$\sqrt{9}=3$$ As a first thought, one might say that (-3) squared is also 9. This is true and (-3) is also the square root of 9. To avoid confusion, if we want the positive square root of a number, also known as the "principal square root", we simply write the number under the square root symbol. When we want the negative square root of the number, we write a negative outside of the square root symbol. For example: $$\sqrt{9}=3$$ $$-\sqrt{9}=-3$$ As we move forward in math, we will often see this process condensed into one line. $$\pm \sqrt{9}=\pm 3$$ This reads as: plus or minus the square root of 9 is equal to plus or minus 3.
Let's look at a few examples.
Example 1: Simplify each. $$\sqrt{64}$$ To find the principal or positive square root of 64, think about what positive number multiplies by itself to give us 64. In the majority of cases, we will find this answer using a calculator. If we don't have a calculator handy, we can factor the number and find the answer:
64 = 2 • 2 • 2 • 2 • 2 • 2
Place the factors in two equal groups:
64 = (2 • 2 • 2) • (2 • 2 • 2)
64 = 8 • 8
Since 8 squared is 64, the square root of 64 is 8. $$\sqrt{64}=8$$ Example 2: Simplify each. $$-\sqrt{121}$$ In this case, we want the negative square root of 121. We want to think about what negative number multiplies by itself to give us 121.
121 = -11 • -11
Since 121 is (-11) squared, we can say that (-11) is the negative square root of 121. $$-\sqrt{(121)}=-11$$
Radical - Refers to the entire expression
Radical Symbol - Refers to the symbol
Radicand - This is the number that is underneath the radical symbol, we may also refer to this as the argument of the radical.
Index - This is the smaller number that is placed at the top left of the radical symbol. The index tells us what root we are asked to find. In other words, an index of 3, would be asking for the cube root. This would be a number that multiplies by itself three times to produce the radicand. An index of 4, would be asking for the fourth root. This would be a number that multiplies by itself four times to produce the radicand. When we have an index of 2, the number is omitted. It is understood that if the index is blank, we are dealing with a square root. For higher-level roots, we always display the index.
If we have the nth root of some real number a: $$\sqrt[n]{a}$$ If n is even and a ≥ 0:
There are two roots:
The Principal nth root of a: $$\sqrt[n]{a}$$ The Negative nth root of a: $$-\sqrt[n]{a}$$ If n is even and a < 0:
The nth root of a is not a real number. This is because an even number of negative factors will always yield a positive product.
If n is odd, then there is one nth root of a: $$\sqrt[n]{a}$$ Let's look at a few examples.
Example 3: Simplify each. $$\sqrt[4]{(625)}$$ We have an index of 4 and a radicand of 625. This means we want to find a number that multiplies by itself four times to produce 625. In other words, what number raised to the fourth power gives us 625. Again, we can begin by factoring 625:
625 = 5 • 5 • 5 • 5
Since 54 is 625, we can say that the fourth root of 625 is 5: $$\sqrt[4]{(625)}=5$$ Example 4: Simplify each. $$\sqrt{(-81)}$$ When we see a negative radicand and an even index, we can stop and write the answer as "no real solution". To think about why this happens, ask yourself the following question. What number multiplied by itself gives us (-81)? The number does not exist because two negatives multiply together to give us a positive. Note: We will learn an alternative answer for this scenario in our upcoming lesson on complex numbers.
Example 5: Simplify each. $$\sqrt[3]{(-64)}$$ When we have a negative radicand and an odd index, we have exactly one root. Let's factor 64:
64 = 2 • 2 • 2 • 2 • 2 • 2
Since our index is a 3, we want to find a number that multiplies by itself 3 times to give us the radicand of 64. We can split the factors up into three equal groups:
64 = (2 • 2) • (2 • 2) • (2 • 2)
We can see that 4 cubed or 4 • 4 • 4 gives us 64. This means the cube root (index of 3) of 64 is 4. Since we have a negative radicand, we can just state our answer as (-4). $$\sqrt[3]{(-64)}=-4$$ Let's discuss one final rule. $$\sqrt[n]{a^n}$$ If n is an even positive integer: $$\sqrt[n]{a^n}=|a|$$ If n is an odd positive integer: $$\sqrt[n]{a^n}=a$$ Let's look at a few examples.
Example 6: Simplify each. $$\sqrt[7]{12^7}$$ Since we have the 7th root of 12 to the 7th power, we are just left with the 12. $$\sqrt[7]{12^7}=12$$ Example 7: Simplify each. $$\sqrt[3]{(-14)^3}$$ Since we have the cube root (index of 3) of (-14) cubed (raised to the 3rd power), we are just left with -14. $$\sqrt[3]{(-14)^3}=-14$$
We will often have to work with "rational exponents", otherwise known as "fractional exponents". $$a^{\frac{1}{n}}=\sqrt[n]{a}$$ $$a^{\frac{m}{n}}=(\sqrt[n]{a})^m$$ Let's look at a few examples.
Example 8: Simplify each. $$16^{\frac{1}{2}}$$ Using the rule above, we can convert 16 raised to the one-half power into the square root of 16. $$16^{\frac{1}{2}}=\sqrt{16}=4$$ Example 9: Simplify each. $$243^{\frac{1}{5}}$$ Using the rule above, we can convert 243 raised to the one-fifth power into the fifth root of 243. $$243^{\frac{1}{5}}=\sqrt[5]{243}=3$$ Example 10: Simplify each. $$1296^{\frac{3}{4}}$$ Using the rule above, we can rewrite this problem as: $$1296^{\frac{3}{4}}=(\sqrt[4]{1296})^3$$ We will first find the fourth root of 1296, which is 6. $$(\sqrt[4]{1296})^3=6^3$$ Now we will raise 6 to the 3rd power. $$6^3=216$$ $$1296^{\frac{3}{4}}=216$$ Example 11: Simplify each, assume all variables represent positive real numbers. $$\large{\left(\frac{4^{-\frac{1}{2}}x^{-\frac{3}{4}}y^{-\frac{5}{8}}}{x^{\frac{3}{4}}y^{-\frac{1}{2}}}\right)^{-3}}$$ To simplify, let's begin by raising each number or variable inside of the parentheses to the power of (-3): $$\large{\frac{4^{\frac{3}{2}}x^{\frac{9}{4}}y^{\frac{15}{8}}}{x^{-\frac{9}{4}}y^{\frac{3}{2}}}}$$ We can raise 4 to the power of 3/2, by taking the square root of 4, and then raising the result of 2 to the power of 3. The result is 8. $$\large{\frac{8x^{\frac{9}{4}}y^{\frac{15}{8}}}{x^{-\frac{9}{4}}y^{\frac{3}{2}}}}$$ To simplify our variables, we use the quotient rule for exponents. We will subtract the exponent in the denominator away from the exponent in the numerator and keep the base the same. If we look at the exponents on the variable x: $$\frac{9}{4}- \left(-\frac{9}{4}\right)=\frac{18}{4}=\frac{9}{2}$$ $$\large{\frac{8x^{\frac{9}{2}}y^{\frac{15}{8}}}{y^{\frac{3}{2}}}}$$ Now we will look at the exponents on the variable y: $$\frac{15}{8}- \frac{3}{2}=\frac{15}{8}- \frac{12}{8}=\frac{3}{8}$$ $$\large{8x^{\frac{9}{2}}y^{\frac{3}{8}}}$$
Let's look at a few examples.
Example 1: Simplify each. $$\sqrt{64}$$ To find the principal or positive square root of 64, think about what positive number multiplies by itself to give us 64. In the majority of cases, we will find this answer using a calculator. If we don't have a calculator handy, we can factor the number and find the answer:
64 = 2 • 2 • 2 • 2 • 2 • 2
Place the factors in two equal groups:
64 = (2 • 2 • 2) • (2 • 2 • 2)
64 = 8 • 8
Since 8 squared is 64, the square root of 64 is 8. $$\sqrt{64}=8$$ Example 2: Simplify each. $$-\sqrt{121}$$ In this case, we want the negative square root of 121. We want to think about what negative number multiplies by itself to give us 121.
121 = -11 • -11
Since 121 is (-11) squared, we can say that (-11) is the negative square root of 121. $$-\sqrt{(121)}=-11$$
nth Root
We are not limited to square roots. When we work with roots, we have to learn some basic terminology:Radical - Refers to the entire expression
Radical Symbol - Refers to the symbol
Radicand - This is the number that is underneath the radical symbol, we may also refer to this as the argument of the radical.
Index - This is the smaller number that is placed at the top left of the radical symbol. The index tells us what root we are asked to find. In other words, an index of 3, would be asking for the cube root. This would be a number that multiplies by itself three times to produce the radicand. An index of 4, would be asking for the fourth root. This would be a number that multiplies by itself four times to produce the radicand. When we have an index of 2, the number is omitted. It is understood that if the index is blank, we are dealing with a square root. For higher-level roots, we always display the index.
If we have the nth root of some real number a: $$\sqrt[n]{a}$$ If n is even and a ≥ 0: There are two roots:
The Principal nth root of a: $$\sqrt[n]{a}$$ The Negative nth root of a: $$-\sqrt[n]{a}$$ If n is even and a < 0:
The nth root of a is not a real number. This is because an even number of negative factors will always yield a positive product.
If n is odd, then there is one nth root of a: $$\sqrt[n]{a}$$ Let's look at a few examples.
Example 3: Simplify each. $$\sqrt[4]{(625)}$$ We have an index of 4 and a radicand of 625. This means we want to find a number that multiplies by itself four times to produce 625. In other words, what number raised to the fourth power gives us 625. Again, we can begin by factoring 625:
625 = 5 • 5 • 5 • 5
Since 54 is 625, we can say that the fourth root of 625 is 5: $$\sqrt[4]{(625)}=5$$ Example 4: Simplify each. $$\sqrt{(-81)}$$ When we see a negative radicand and an even index, we can stop and write the answer as "no real solution". To think about why this happens, ask yourself the following question. What number multiplied by itself gives us (-81)? The number does not exist because two negatives multiply together to give us a positive. Note: We will learn an alternative answer for this scenario in our upcoming lesson on complex numbers.
Example 5: Simplify each. $$\sqrt[3]{(-64)}$$ When we have a negative radicand and an odd index, we have exactly one root. Let's factor 64:
64 = 2 • 2 • 2 • 2 • 2 • 2
Since our index is a 3, we want to find a number that multiplies by itself 3 times to give us the radicand of 64. We can split the factors up into three equal groups:
64 = (2 • 2) • (2 • 2) • (2 • 2)
We can see that 4 cubed or 4 • 4 • 4 gives us 64. This means the cube root (index of 3) of 64 is 4. Since we have a negative radicand, we can just state our answer as (-4). $$\sqrt[3]{(-64)}=-4$$ Let's discuss one final rule. $$\sqrt[n]{a^n}$$ If n is an even positive integer: $$\sqrt[n]{a^n}=|a|$$ If n is an odd positive integer: $$\sqrt[n]{a^n}=a$$ Let's look at a few examples.
Example 6: Simplify each. $$\sqrt[7]{12^7}$$ Since we have the 7th root of 12 to the 7th power, we are just left with the 12. $$\sqrt[7]{12^7}=12$$ Example 7: Simplify each. $$\sqrt[3]{(-14)^3}$$ Since we have the cube root (index of 3) of (-14) cubed (raised to the 3rd power), we are just left with -14. $$\sqrt[3]{(-14)^3}=-14$$
Rational Exponents
We will often have to work with "rational exponents", otherwise known as "fractional exponents". $$a^{\frac{1}{n}}=\sqrt[n]{a}$$ $$a^{\frac{m}{n}}=(\sqrt[n]{a})^m$$ Let's look at a few examples.
Example 8: Simplify each. $$16^{\frac{1}{2}}$$ Using the rule above, we can convert 16 raised to the one-half power into the square root of 16. $$16^{\frac{1}{2}}=\sqrt{16}=4$$ Example 9: Simplify each. $$243^{\frac{1}{5}}$$ Using the rule above, we can convert 243 raised to the one-fifth power into the fifth root of 243. $$243^{\frac{1}{5}}=\sqrt[5]{243}=3$$ Example 10: Simplify each. $$1296^{\frac{3}{4}}$$ Using the rule above, we can rewrite this problem as: $$1296^{\frac{3}{4}}=(\sqrt[4]{1296})^3$$ We will first find the fourth root of 1296, which is 6. $$(\sqrt[4]{1296})^3=6^3$$ Now we will raise 6 to the 3rd power. $$6^3=216$$ $$1296^{\frac{3}{4}}=216$$ Example 11: Simplify each, assume all variables represent positive real numbers. $$\large{\left(\frac{4^{-\frac{1}{2}}x^{-\frac{3}{4}}y^{-\frac{5}{8}}}{x^{\frac{3}{4}}y^{-\frac{1}{2}}}\right)^{-3}}$$ To simplify, let's begin by raising each number or variable inside of the parentheses to the power of (-3): $$\large{\frac{4^{\frac{3}{2}}x^{\frac{9}{4}}y^{\frac{15}{8}}}{x^{-\frac{9}{4}}y^{\frac{3}{2}}}}$$ We can raise 4 to the power of 3/2, by taking the square root of 4, and then raising the result of 2 to the power of 3. The result is 8. $$\large{\frac{8x^{\frac{9}{4}}y^{\frac{15}{8}}}{x^{-\frac{9}{4}}y^{\frac{3}{2}}}}$$ To simplify our variables, we use the quotient rule for exponents. We will subtract the exponent in the denominator away from the exponent in the numerator and keep the base the same. If we look at the exponents on the variable x: $$\frac{9}{4}- \left(-\frac{9}{4}\right)=\frac{18}{4}=\frac{9}{2}$$ $$\large{\frac{8x^{\frac{9}{2}}y^{\frac{15}{8}}}{y^{\frac{3}{2}}}}$$ Now we will look at the exponents on the variable y: $$\frac{15}{8}- \frac{3}{2}=\frac{15}{8}- \frac{12}{8}=\frac{3}{8}$$ $$\large{8x^{\frac{9}{2}}y^{\frac{3}{8}}}$$
Skills Check:
Example #1
Simplify each. $$\sqrt[3]{-1{,}331}$$
Please choose the best answer.
A
$$-15$$
B
$$-9$$
C
$$-19$$
D
$$-11$$
E
$$-13$$
Example #2
Simplify each, assume all variables represent positive real numbers. $$\left(64x^{4}\right)^{-\frac{1}{2}}$$
Please choose the best answer.
A
$$x^{4}$$
B
$$64x^{6}$$
C
$$\frac{1}{8x^{2}}$$
D
$$\frac{1}{4x}$$
E
$$\frac{1}{16x^{8}}$$
Example #3
Simplify each, assume all variables represent positive real numbers. $$\frac{\left(x^{\frac{1}{2}}\right)^{2}}{x^{-2}\cdot x}$$
Please choose the best answer.
A
$$x^{6}$$
B
$$x^{2}$$
C
$$\frac{1}{x^{2}}$$
D
$$\frac{1}{x^{4}}$$
E
$$x^{8}$$
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