ISEE Upper Level

Radical & Rational Expressions

Q: Why can't the bottom of a fraction be zero?

Imagine having `5` pizzas and trying to share them with `0` people. It doesn't make any sense! That's why dividing by zero is mathematically impossible.

Simplifying radical expressions, rationalizing denominators, simplifying rational expressions, and domain restrictions — non-polynomial expression manipulation

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Imagine you're playing a video game where some characters are trapped in an escape room, and others are stuck trying to balance on a giant seesaw! 🎮 In algebra, we have our own versions of these puzzles called Radicals and Rational Expressions.

First, let's talk about the escape room, which we call a "radical" (it looks like this: $x$ ). To escape the square root room, numbers need to find their identical twin! If the number $25$ is in the room, it breaks into $5 \cdot 5$ . Because they are twins, a $5$ gets to escape! 🏃‍♂️

Next, we have the seesaw, which is a "rational expression." That's just a fancy math word for a giant fraction with letters (variables) mixed in. Just like you can simplify the fraction $\frac{2}{4}$ to $\frac{1}{2}$ , you can shrink down giant math fractions by crossing out matching pieces on the top and the bottom. 🍕

But there is one super important rule for these giant fractions: The bottom can NEVER be zero. It's like the floor is lava! 🌋 If the bottom becomes zero, the math universe explodes. On the ISEE, you will be a math detective looking for the secret numbers that make the bottom zero so you can avoid them. Let's learn how to conquer these awesome math puzzles!

Key Formula

The Floor is Lava Rule: If you have a rational expression like

\frac{a}{x - b}

, then

x

can NEVER equal

b

, because

\frac{a}{0}

is impossible!

Practice Questions

3 practice questions for ISEE Upper Level

Q1 Hard

Which expression is equivalent to the expression

25 y^{36}

5 y^{6}

5 y^{18}

12.5 y^{18}

25 y^{6}

Show Solution

To simplify the expression $25 y^{36}$ , take the square root of the coefficient and the variable separately. The square root of $25$ is $5$ . To find the square root of $y^{36}$ , divide the exponent by $2$ , which gives $y^{18}$ . Combining these results yields $5 y^{18}$ .

Answer: B

Q2 Hard

Which expression is equivalent to

\frac{81 a ^{4}}{4 b ^{10}}

for all positive values of

a

and

b

\frac{9 a ^{2}}{2 b ^{5}}

\frac{9 a ^{2}}{4 b ^{5}}

\frac{40.5 a ^{2}}{2 b ^{5}}

\frac{81 a ^{2}}{4 b ^{5}}

Show Solution

To simplify the radical expression $\frac{81 a ^{4}}{4 b ^{10}}$ , you can take the square root of the numerator and the denominator separately. The square root of the numerator, $81 a^{4}$ , is $9 a^{2}$ . The square root of the denominator, $4 b^{10}$ , is $2 b^{5}$ . Placing the simplified numerator over the simplified denominator gives $\frac{9 a ^{2}}{2 b ^{5}}$ .

Answer: A

Q3 Hard

Which expression is equivalent to

\frac{36 x ^{8}}{9 x ^{2}}

for all

x > 0

2 x^{3}

2 x^{4}

4 x^{3}

4 x^{4}

Show Solution

There are two ways to solve this. Method 1: Combine the expression under a single radical first: $\frac{36 x ^{8}}{9 x ^{2}}$ . Simplify the fraction inside to get $4 x^{6}$ . Taking the square root gives $2 x^{3}$ .

Method 2: Simplify the radical expressions in the numerator and the denominator first. The square root of $36 x^{8}$ is $6 x^{4}$ . The square root of $9 x^{2}$ is $3 x$ . Now substitute these back into the fraction to get $\frac{6 x ^{4}}{3 x}$ . Divide the coefficients ( $6 \div 3 = 2$ ) and subtract the exponents of the variables ( $x^{4} \div x = x^{4 - 1} = x^{3}$ ) to get $2 x^{3}$ .

Answer: A

Tips & Strategies

On the ISEE, if you see a giant fraction, always check the bottom first! Look at the answer choices to see what $x$ cannot be before you even start doing the math. 🕵️‍♀️
When breaking numbers out of a square root room, write out their factors. Look for perfect squares like $4$ , $9$ , $16$ , or $25$ to make the escape much faster!
Never just cross out pieces of an addition problem! In $\frac{x + 2}{2}$ , you CANNOT cross out the $2$ s. You can only cross out things that are being multiplied.

Common Mistakes

Watch out for the 'illegal crossing out' trap! 🪤 Students often try to simplify $\frac{x + 4}{4}$ by crossing out the $4$ s to get $x$ . You can only cancel numbers that are multiplied together, not added!
Don't forget that $16$ is just $4$ , not $4$ . Once the number escapes the radical room, the radical sign goes away completely!

Frequently Asked Questions

What does 'rational' even mean?

In math, 'rational' just means 'fraction'. The word comes from 'ratio', which is a way to compare two numbers, like slices of pizza to whole pizzas! 🍕

Why can't the bottom of a fraction be zero?

Imagine having $5$ pizzas and trying to share them with $0$ people. It doesn't make any sense! That's why dividing by zero is mathematically impossible.

How does the ISEE test this topic?

The ISEE might ask you to simplify an expression or ask 'for what value of x is this expression undefined?' Undefined is just a fancy test-maker word for 'the bottom is zero'.

Will I get points taken off if I guess wrong?

Nope! The ISEE has no guessing penalty. 🎯 If a radical or rational expression looks too scary, take your best guess and move on to a question you feel great about.

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