# 1.3 Radicals and rational exponents  (Page 2/11)

 Page 2 / 11

## 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.
3. Simplify.

## Using the product rule to simplify square roots

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

$\sqrt{12}\cdot \sqrt{3}$

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

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

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

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

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.

can you solve it step b step
what is linear equation with one unknown 2x+5=3
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2