Lecture: Radicals, Rational Exponents & Radical Equations
Lecture script · about 15–20 minutes spoken · Week 12
Opening
A landscaper knows a garden's area and needs the length of its sides. A safety engineer wants to know how long a dropped tool takes to reach the ground. A nurse or a biologist meets formulas with fractional powers in them. All three are running a power backwards. That move has a name, a root, and it is the subject of this lecture.
Here's where we are in this course. In Week 1 you learned the exponent rules and the principal square root, and you met the powers of two, from 1 up to 1024. In Week 3 you learned that the expression under a square root, the radicand, can't be negative. In Week 6 you squared binomials, and in Week 7 you solved quadratic equations. Last week, in Week 11, you saw that a correct solving step can still hand you a candidate that fails the original equation. All of that comes back in this lecture. Next week, in Week 13, the variable moves up into the exponent itself, and your exponent rules come with it.
I'll take eight ideas in order, from what a root is to the check that catches a false answer.
1. Read a root as the undo of a power
Adding 5 is undone by subtracting 5. A root is the move that undoes a power.
Picture a square vegetable garden that covers 64 square feet. How long is each side? You need a number that, times itself, gives 64. That's 8, because 8 times 8 is 64. So the square root of 64 is 8, and each side is 8 feet.
Here are the words for that. A radical is an expression that uses a root sign, such as the square root of 50 or the cube root of 8, to name the number whose power gives the value under the sign. The small number tucked into the crook of the root sign is the index. It says which power the root undoes. A square root has index 2, and we usually don't write it. The nth root of a is the number whose nth power is a. For an even index, it's the nonnegative one, and the number under the sign must be 0 or more.
Week 1's powers of two turn into a root table when you read them backwards. Since 4 squared is 16, the square root of 16 is 4. Since 32 squared is 1024, the square root of 1024 is 32. Since 2 cubed is 8, the cube root of 8 is 2. And since 2 to the fourth is 16, the fourth root of 16 is 2.
Pause and try this. Is the fourth root of 16 equal to 4, because 16 divided by 4 is 4, or equal to 2?
Hold your answer for a moment, then read on.
Here's the answer. It's 2, because 2 times 2 times 2 times 2 is 16, while 4 used four times as a factor is 256, not 16.
2. Simplify a square root with the product rule
Roots have a friendly rule for products. The product rule for radicals says that for a and b that are 0 or more, the square root of a times b equals the square root of a times the square root of b. Try it on 9 times 16. That product is 144, and the square root of 144 is 12. The square root of 9 is 3, the square root of 16 is 4, and 3 times 4 is 12. Same answer either way.
That rule lets us tidy up a square root. A perfect square is a number that is the square of a whole number, such as 16, 25, or 64. To simplify, you pull the largest perfect-square factor out from under the root.
Take the square root of 50. The largest perfect square that divides 50 is 25, because 50 is 25 times 2. So the square root of 50 is the square root of 25 times the square root of 2, which is 5 times the square root of 2. Check it by squaring: 5 squared is 25, the square root of 2 squared is 2, and 25 times 2 is 50. As a decimal, the square root of 2 is about 1.4142, so 5 times the square root of 2 is about 7.071. The exact form is the answer, and the decimal is your check.
Now the square root of 48. Here's a common stumble. Many people write 48 as 4 times 12 and get 2 times the square root of 12. But 12 still holds the perfect square 4. So 2 times the square root of 12 becomes 2 times 2 times the square root of 3, which is 4 times the square root of 3. Start with the largest perfect square, 16, and you get 4 times the square root of 3 in one step. That finished shape is called simplified radical form: no perfect-square factor other than 1 is left under the root sign.
Pause and try this. Is 2 times the square root of 12 in simplified radical form?
Think it through before the answer arrives.
Here's the answer. No, because 12 still contains the perfect square 4, so it simplifies further to 4 times the square root of 3.
3. Use the quotient rule: the dropped object
Fractions get a matching rule. The quotient rule for radicals says that for a that is 0 or more and b greater than 0, the square root of a over b equals the square root of a over the square root of b.
Here's where it earns its keep. Remember the launched object from Week 7, fired straight up from an 80-foot platform. It peaks at 144 feet, 2 seconds after launch, and lands at 5 seconds. An object dropped from rest falls h feet in a time equal to the square root of h over 16. That 16 is the same 16 as in the launched object's height formula, half of gravity's pull of 32 feet per second per second. Like that formula, it ignores air resistance.
So how long does the fall from the peak take? The time is the square root of 144 over 16. By the quotient rule, that's the square root of 144, which is 12, over the square root of 16, which is 4. And 12 over 4 is 3 seconds. You can also divide first: 144 over 16 is 9, and the square root of 9 is 3.
Notice the match. The launched object peaks at 2 seconds and lands at 5 seconds, and 5 minus 2 is 3 seconds of falling. Two different formulas, one with a root and one with a square, agree on the same 3 seconds.
Pause and try this. Someone takes the root of only the bottom and writes 144 over 4. What do they get, and is it right?
Give it a moment.
Here's the answer. They get 36, which is wrong, because the root sign covers the whole fraction, so the true value is 3.
4. Never split a root over a sum
It's tempting to invent a third rule, for sums. There isn't one.
Take the square root of 9 plus 16. Add inside first: 9 plus 16 is 25, and the square root of 25 is 5. Now split it the wrong way: the square root of 9 is 3, the square root of 16 is 4, and 3 plus 4 is 7. So 5 and 7 aren't equal. One counterexample is enough to sink a rule.
Here's a way to hold onto it. The bar on top of a root sign works like a pair of parentheses. Under the Week 1 order of operations, you finish everything inside the grouping first. So you add under the bar, and only then take the root. Roots split over a product, like 9 times 16. They never split over a sum.
Pause and try this. Which one is 12: the square root of 9 times 16, or the square root of 9 plus 16?
Say which one before you read on.
Here's the answer. The square root of 9 times 16 is 12, because roots split over a product; the square root of 9 plus 16 is 5.
5. Read a rational exponent as a root
Exponents can be fractions. Ask what that must mean for the Week 1 rules to keep working.
The power rule says a power raised to a power multiplies the exponents. So take a to the one-half, and square it. One-half times 2 is 1, so you get a to the first, which is a. That means a to the one-half is a number whose square is a. That's the square root of a. The rules themselves force the meaning.
Here's the general term. A rational exponent is an exponent that is a fraction, such as the two-thirds in 8 to the two-thirds. The denominator names the root and the numerator names the power, so a to the m over n means the nth root of a, raised to the power m.
Let's do 8 to the two-thirds. The bottom number, 3, says cube root. The top number, 2, says square. Take the root first: the cube root of 8 is 2. Then square it: 2 squared is 4. So 8 to the two-thirds is 4. Next, 16 to the three-fourths. The fourth root of 16 is 2, and 2 cubed is 8. And 27 to the two-thirds: the cube root of 27 is 3, and 3 squared is 9.
A negative sign in the exponent keeps its Week 1 meaning: it asks for a reciprocal. So 8 to the negative two-thirds is 1 over 8 to the two-thirds, which is 1 over 4. It's one-fourth, not negative 4.
Why root first? If you cube 16 first, you get 4096. Its fourth root is still 8, but you're juggling a much bigger number. Root first keeps the numbers small.
Pause and try this. Someone works out 16 to the three-fourths as 16 times three-fourths, which is 12. What went wrong?
Find the slip before the answer arrives.
Here's the answer. They treated the exponent as a multiplier; an exponent is a root and a power, so the value is 8, not 12.
6. Simplify with rational exponents
Every Week 1 exponent rule still works with fraction exponents, so the new skill is fraction arithmetic, like adding a half and a third of a pizza.
Start with x to the one-half times x to the one-third. Same base, multiplying, so you add the exponents. The common denominator is 6. One-half is three-sixths, one-third is two-sixths, and together they make five-sixths. So the product is x to the five-sixths. Let's check it at 64, a power of two. 64 to the one-half is 8, 64 to the one-third is 4, and 8 times 4 is 32. On the other side, the sixth root of 64 is 2, and 2 to the fifth is 32. They agree.
Next, x to the three-halves, all squared. A power raised to a power, so you multiply: three-halves times 2 is 3, and you get x cubed. Check at 4. Four to the three-halves is the square root of 4, cubed, which is 2 cubed, or 8. And 8 squared is 64, which is also 4 cubed.
Last, x to the five-fourths divided by x to the one-fourth. Same base, dividing, so you subtract: five-fourths minus one-fourth is four-fourths, which is 1. So you get x. Check at 16: 16 to the five-fourths is 32, 16 to the one-fourth is 2, and 32 divided by 2 is 16.
Pause and try this. If someone multiplies the exponents in x to the one-half times x to the one-third, what do they get at 64?
Try it in your head first.
Here's the answer. They get x to the one-sixth, which is 2 at 64, not 32, because a product of powers adds exponents rather than multiplying them.
7. Solve a radical equation
A radical equation is an equation with the variable under a root sign. There's a four-step plan. First, isolate the radical, which means rearranging so the root stands alone on one side. Second, raise both sides to the index, so square both sides for a square root. Third, solve. Fourth, check every candidate in the original equation.
Start easy. The square root of x equals 5. The root is already alone, so square both sides, and x is 25. Check: the square root of 25 is 5. In the garden setting, a square bed with a 5-foot side covers 25 square feet.
Next, the square root of 2x minus 1 equals 3. Square both sides to get 2x minus 1 equals 9. Add 1, so 2x is 10. Divide by 2, so x is 5. Check: 2 times 5 minus 1 is 9, and the square root of 9 is 3.
Now one where isolating matters. The square root of x plus 1, plus 2, equals 6. Subtract 2 from both sides first, so the square root of x plus 1 equals 4. Square both sides: x plus 1 is 16, so x is 15. Check: the square root of 16 is 4, and 4 plus 2 is 6.
And the dropped object in reverse. How high must a drop start for the fall to last 3 seconds? Set 3 equal to the square root of h over 16. Square both sides: 9 equals h over 16. Multiply by 16, and h is 144 feet, the launched object's peak.
Pause and try this. In the square root of x plus 1, plus 2, equals 6, why do you subtract the 2 before squaring?
Give yourself a few seconds here.
Here's the answer. Squaring with the 2 still attached doesn't square the root alone, so you isolate the radical first to keep the equation honest.
8. Check every candidate: extraneous solutions
Squaring is powerful, but it forgets signs. Both 1 and negative 1 square to 1. So squaring can turn a false statement into a true one, and a false candidate can slip through. Last week you met the name for that: an extraneous solution, a candidate that a correct step produced but that fails the original equation.
Watch it happen. Solve the square root of x plus 3 equals x plus 1. The root is alone, so square both sides. On the left you get x plus 3. On the right, use the Week 6 special product: x plus 1, squared, is x squared plus 2x plus 1. Move everything to one side, and you get x squared plus x minus 2 equals 0. That factors as x plus 2 times x minus 1, so the candidates are negative 2 and 1.
Now check each one in the original. At 1, the left side is the square root of 4, which is 2, and the right side is 1 plus 1, which is 2. It works. At negative 2, the left side is the square root of 1, which is 1. But the right side is negative 2 plus 1, which is negative 1. And 1 isn't negative 1, and a principal square root is never negative. So negative 2 is extraneous, and the only solution is 1.
Here's a picture in words. The curve for the square root of x plus 3 starts at the point negative 3, 0, and rises slowly. The line x plus 1 rises more steeply. They cross once, at the point 1, 2, and never at negative 2.
Pause and try this. The square root of x equals negative 3. Squaring gives x equals 9. Is 9 a solution?
Check it before you read on.
Here's the answer. No, because the square root of 9 is 3, not negative 3; a principal square root is never negative, so there's no solution.
Closing
Let's pull it together, in the order we took it. A root undoes a power, and the index says which power. You simplify a square root by pulling out its largest perfect-square factor, so the square root of 50 is 5 times the square root of 2. A root splits over a product or a fraction, which is how the dropped object falls for 3 seconds. It never splits over a sum. A rational exponent is a root: the bottom number is the root, the top number is the power, and you take the root first. The Week 1 rules work on fraction exponents: add for a product, multiply for a power of a power, subtract for a quotient. To solve a radical equation, isolate, square, solve, and check. And because squaring forgets signs, every candidate goes back into the original equation.
Next in this module, you'll find the slide deck, then the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Week 13, Exponential Functions, the variable climbs into the exponent, and the rules you used here carry straight into that unit.
Built with The Course Maker · samples.thecoursemaker.com