10.1: Simplify Radicals
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Not all radicands are perfect squares, where when we take the square root, we obtain a positive integer. For example, if we input √8 in a calculator, the calculator would display 2.828427124746190097603377448419⋯ and even this number is a rounded approximation of the square root. To be as accurate as possible, we will leave all answers in exact form, i.e., answers contain integers and radicals- no decimals.
When we say to simplify an expression with radicals, the simplified expression should have
- a radical, unless the radical reduces to an integer
- a radicand with no factors containing perfect squares
- no decimals
Following these guidelines ensures the expression is in its simplest form.
Simplify Radicals
If a, b are any two positive real numbers, then √ab=√a⋅√b
Simplify: √75
Solution
We can apply the product rule for radicals to simplify this number. We need to find the largest factor of 75 that is a perfect square (since we have a square root) and rewrite the radicand as a product of this perfect square and its other factor. The largest factor of radicand 75 that is a perfect square is 25.
√75Rewrite radicand as a product of 25 and 3√25⋅3Apply product rule for radicals√25⋅√3Simplify each square root5⋅√3Rewrite5√3Simplified expression
If the radicand is not a perfect square, we leave as is; hence, we left √3 as is.
Simplify: √72
Solution
We can apply the product rule for radicals to simplify this number. We need to find the largest factor of 72 that is a perfect square (since we have a square root) and rewrite the radicand as a product of this perfect square and its other factor. The largest factor of radicand 72 that is a perfect square is 36.
√72Rewrite radicand as a product of 36 and 2√36⋅2Apply product rule for radicals√36⋅√2Simplify each square root6⋅√2Rewrite6√2Simplified expression
If the radicand is not a perfect square, we leave as is; hence, we left √2 as is.
Simplify Radicals with Coefficients
Simplify: 5√63
Solution
We can apply the product rule for radicals to simplify this number and multiply coefficients in the last steps. We need to find the largest factor of 63 that is a perfect square (since we have a square root) and rewrite the radicand as a product of this perfect square and its other factor. The largest factor of radicand 63 that is a perfect square is 9.
5√63Rewrite radicand as a product of 9 and 75√9⋅7Apply product rule for radicals5⋅√9⋅√7Simplify each square root5⋅3⋅√7Rewrite and simplify coefficients15√7Simplified expression
If the radicand is not a perfect square, we leave as is; hence, we left √7 as is.
Rational Exponents
When we simplify radicals, we extract roots of factors with exponents in which are multiples of the root (index). For example, √x4=2√x4=x2, but notice we just divided the power on x by the root. Let’s look at the example again, but now as division of exponents:
√x4=3√x4=x42=x2
Division with exponents, or fraction exponents, are called rational exponents.
Let a be the base, and m and n be real real numbers. Then
amn=n√am=(n√a)m
The denominator of a rational exponent is the root on the radical and vice versa
Rewrite each radical with its corresponding rational exponent.
- (5√x)3
- (6√3x)5
- 1(7√a)3
- 1(3√xy)2
Solution
- For the expression (5√x)3, we see the root is 5. This means that the denominator of the rational exponent is 5. Hence, the numerator is the exponent 3: (5√x)3=x35.
- For the expression (6√3x)5, we see the root is 6. This means that the denominator of the rational exponent is 6. Hence, the numerator is the exponent 5: (6√3x)5=(3x)56.
- For the expression 1(7√a)3, we see the root is 7. This means that the denominator of the rational exponent is 7. Hence, the numerator is the exponent 3. Furthermore, since the expression with the radical is in the denominator, we can rewrite the expression using a negative exponent: 1(7√a)3=(a)−37.
- For the expression 1(3√xy)2, we see the root is 3. This means that the denominator of the rational exponent is 3. Hence, the numerator is the exponent 2. Furthermore, since the expression with the radical is in the denominator, we can rewrite the expression using a negative exponent: 1(3√xy)2=(xy)−23.
Rewrite each expression in its equivalent radical form.
- a53
- (2mn)27
- x−45
- (xy)−29
Solution
- From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: a53=3√a5 or (3√a)5.
- From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: (2mn)27=7√(2mn)2 or (7√2mn)2.
- From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: x−45=(5√x)−4. Notice that the expression still contains a negative exponent. Hence, we need to reciprocate the radical to rewrite the expression with only positive exponents: x−45=1(5√x)4
- From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: (xy)−29=(9√x)−2. Notice that the expression still contains a negative exponent. Hence, we need to reciprocate the radical to rewrite the expression with only positive exponents: (xy)−29=1(9√xy)2
Nicole Oresme, a Mathematician born in Normandy was the first to use rational exponents. He used the notation 13•9p to represent 913. However, his notation went largely unnoticed
The ability to change between rational exponential expressions and radical expressions allows us to evaluate expressions.
Evaluate 27−43.
Solution
We first rewrite the expression with only positive exponents, then evaluate the exponen
27−43Rewrite the expression with positive exponents12743Rewrite in radical form1(3√27)4Evaluate radical 3√27=31(3)4Evaluate exponent 34=81181Result
Thus, 27−43=181. This result should emphasize the fact that negative exponents means reciprocals, and not negative numbers.
Simplify Radicals with Variables
Commonly, radicands can contain variables. When taking the square roots of variables, we know the root is 2; we do not always write it, but we know it’s there. Hence, we apply the product rule of radicals by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.
Simplify: √x6y5
Solution
We can apply the product rule for radicals to simplify by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.
√x6y5Rewrite radicand√x6⋅y4⋅y1Apply product rule for radicals√x6⋅√y4⋅√ySimplify each square rootx3⋅y2⋅√yRewrite and simplify coefficientsx3y2√ySimplified expression
Notice that (x3)2 and (y2)2=y4; hence, we extract the perfect squares of the variables and leave the √y as is.
Recall, when taking a square root of a number, the radicand must be greater than or equal to zero. So, when we are applying the square root to variables, the variables must also be greater than or equal to zero.
Notice, we are essentially dividing the exponents on the variables by two and the factor that remains in the radicand has exponent 1.
Simplify: −5√18x4y6z10. Assume all variables are positive.
Solution
We can apply the product rule for radicals to simplify by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.
−5√18x4y6z10Rewrite radicand−5⋅√9⋅2⋅x4⋅y6⋅x10Apply product rule for radicals−5⋅√9⋅√2⋅√x4⋅√y6⋅√z10Simplify each square root−5⋅3⋅√2⋅x2⋅y3⋅z5Rewrite and simplify coefficients−15x2y3z5√2Simplified expression
Simplify: √20x5y9z6. Assume all variables are positive.
Solution
We can apply the product rule for radicals to simplify by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.
√20x5y9z6Rewrite radicand√4⋅5⋅x4⋅x⋅y8⋅y⋅z6Apply product rule for radicals√4⋅√5⋅√x4⋅√x⋅√y8⋅√y⋅√z6Simplify each square root2⋅√5⋅x2⋅√x⋅y4⋅√y⋅z3Rewrite and simplify coefficients2x2y4z3√5xySimplified expression
Simplify Radicals Homework
Simplify. Assume all variables are positive.
√245
√36
√12
3√12
6√128
−8√392
√192n
√196v2
√252x2
−√100k4
−7√64x4
−5√36m
√45x2y2
√16x3y3
√320x4y4
6√80xy2
5√245x2y3
−2√180u3v
−8√180x4y2z4
2√80hj4k
−4√54mnp2
√125
√196
√338
5√32
7√128
−7√63
√343b
√100n3
√200a3
−4√175p4
−2√128n
8√112p2
√72a3b4
√512a4b2
√512m4n3
8√98mn
2√72x2y2
−5√72x3y4
6√50a4bc2
−√32xy2z3
−8√32m2p4q
Write each expression in radical form with only positive exponents.
m35
(7x)32
(10r)−34
(6b)−43
Write each expression in exponential form.
1(√6x)3
1(4√n)7
√v
√5a
Evaluate without using a calculator.
823
432
1614
100−32