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### Course: Algebra 1>Unit 11

Lesson 3: Simplifying square roots

## What are exponent properties?

We can use these properties of exponents to help us simplify expressions involving exponents:
Product rule: ${x}^{a}×{x}^{b}={x}^{a+b}$. For example, ${x}^{2}×{x}^{3}={x}^{5}$.
Power rule: $\left({x}^{a}{\right)}^{b}={x}^{ab}$. For example, $\left({x}^{2}{\right)}^{3}={x}^{6}$.
Quotient rule: $\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}$. For example, $\frac{{x}^{5}}{{x}^{2}}={x}^{3}$.
Zero exponent rule: ${x}^{0}=1$. For example, ${7}^{0}=1$.

A radical is a symbol that we use to write square roots, cube roots, and other roots. For example, $\sqrt{81}$ is the square root of $81$, or the number we can multiply by itself to get $81$. Another example is $\sqrt[3]{\phantom{A}8}$, which is the cube root of $8$, or the number we can multiply by itself to get $8$.
Practice with our Square roots exercise.
Practice with our Cube roots exercise.

## What are some common ways to simplify square roots?

One common way to simplify square roots is to factor the radicand (the number inside the square root symbol) into perfect squares. For example, $\sqrt{50}$ can be simplified by factoring $50$ into $25$ times $2$:
$\sqrt{50}=\sqrt{25×2}=\sqrt{25}×\sqrt{2}=5\sqrt{2}$

## Want to join the conversation?

• There's a mistake on this page where it says "the cubed root of 8 is the number we can multiply by itself to get 8". It should actually say the number we can multiply by itself 3 times to get 8 (because it's the cubed root, not the square root)
• help me I have a high school entrance exam soon, what should I study to get a good score?
• Thank you guys, all of these answers are helping me
• Is this correct?

Product: if you multiply a number with an exponent by the same number but with a different exponent, you add the two exponents together, leaving the whole number as is.

Power: If you multiply a number with an exponent by an exponent, you multiply the two exponents and leave the base number alone.

Quotient: Since division is the opposite of multiplication, naturally the addition power of multiplying exponents becomes subtraction when you divide them. (Be careful with negative numbers! They have tripped me up several times.)

Zero exponent rule: strange and tricky. imagine it like this: you have a number you want to multiply exponentially, say, 7. Instead of thinking this as the base, think of the neutral base as 1. If 1 is multiplied exponentially by 7 zero times, it wasn't multiplied at all. so it remains its original self: 1. If it was multiplied once, it would be 7. (This is just an idea of how to think about it! please someone comment if you see a flaw in my logic.)

• All that is correct! I like how you describe each one in your own words. It shows a true understanding of the concept!
• What's a rational number?
(1 vote)
• @EllieValeska223

Great Question! A rational number is any number that can be written as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers, and the denominator is not equal to zero. In other words, a rational number can be expressed as p/q, where p and q are both integers and q ≠ 0.

Hope this helps.
• Can I find out which perfect squares I use to fully simplify and replace the radicand? For example, the square root of 160 could be simplified like 2 and the square root of 40, but the fully simplified answer is 4 to the square root of 10.
(1 vote)
• The square root of 40 isn't simplified since 40 = 4 x 10, with the 4 being a perfect square.
(1 vote)
• Is this a chat room?
(1 vote)
• hey.... how can i tell the different rules apart?
(1 vote)
• Why do we do this