1.3 Radicals and rational exponents  (Page 2/11)

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The product rule for simplifying square roots

If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the product $\text{\hspace{0.17em}}ab\text{\hspace{0.17em}}$ is equal to the product of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$

$\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$

Given a square root radical expression, use the product rule to simplify it.

1. Factor any perfect squares from the radicand.
2. Write the radical expression as a product of radical expressions.
3. Simplify.

Using the product rule to simplify square roots

Simplify the radical expression.

1. $\sqrt{300}$
2. $\sqrt{162{a}^{5}{b}^{4}}$

Simplify $\text{\hspace{0.17em}}\sqrt{50{x}^{2}{y}^{3}z}.$

$5|x||y|\sqrt{2yz}.\text{\hspace{0.17em}}$ Notice the absolute value signs around x and y ? That’s because their value must be positive!

Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

1. Express the product of multiple radical expressions as a single radical expression.
2. Simplify.

Using the product rule to simplify the product of multiple square roots

Simplify the radical expression.
$\sqrt{12}\cdot \sqrt{3}$

Simplify $\text{\hspace{0.17em}}\sqrt{50x}\cdot \sqrt{2x}\text{\hspace{0.17em}}$ assuming $\text{\hspace{0.17em}}x>0.$

$10|x|$

Using the quotient rule to simplify square roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite $\text{\hspace{0.17em}}\sqrt{\frac{5}{2}}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}\frac{\sqrt{5}}{\sqrt{2}}.$

The quotient rule for simplifying square roots

The square root of the quotient $\text{\hspace{0.17em}}\frac{a}{b}\text{\hspace{0.17em}}$ is equal to the quotient of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b,$ where $\text{\hspace{0.17em}}b\ne 0.$

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Given a radical expression, use the quotient rule to simplify it.

1. Write the radical expression as the quotient of two radical expressions.
2. Simplify the numerator and denominator.

Using the quotient rule to simplify square roots

Simplify the radical expression.

$\sqrt{\frac{5}{36}}$

Simplify $\text{\hspace{0.17em}}\sqrt{\frac{2{x}^{2}}{9{y}^{4}}}.$

$\frac{x\sqrt{2}}{3{y}^{2}}.\text{\hspace{0.17em}}$ We do not need the absolute value signs for $\text{\hspace{0.17em}}{y}^{2}\text{\hspace{0.17em}}$ because that term will always be nonnegative.

Using the quotient rule to simplify an expression with two square roots

Simplify the radical expression.

$\frac{\sqrt{234{x}^{11}y}}{\sqrt{26{x}^{7}y}}$

Simplify $\text{\hspace{0.17em}}\frac{\sqrt{9{a}^{5}{b}^{14}}}{\sqrt{3{a}^{4}{b}^{5}}}.$

${b}^{4}\sqrt{3ab}$

Adding and subtracting square roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of $\text{\hspace{0.17em}}\sqrt{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}3\sqrt{2}\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}4\sqrt{2}.\text{\hspace{0.17em}}$ However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression $\text{\hspace{0.17em}}\sqrt{18}\text{\hspace{0.17em}}$ can be written with a $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ in the radicand, as $\text{\hspace{0.17em}}3\sqrt{2},$ so $\text{\hspace{0.17em}}\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}.$

Given a radical expression requiring addition or subtraction of square roots, solve.

1. Simplify each radical expression.
2. Add or subtract expressions with equal radicands.

Questions & Answers

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
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Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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