Properties Of Square Roots - Varsity Tutors

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Properties of Square Roots

Study Guide

Key Definition

For all positive real numbers $a$ and $b$, the product property is: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and the quotient property is: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ where $b \neq 0$.

Important Notes

  • The product property of square roots can simplify multiplication.
  • The quotient property of square roots can simplify division.
  • Square roots are defined for non-negative real numbers in these properties.
  • These properties do not apply to negative numbers under the square root.
  • When simplifying variable expressions like $\sqrt{a^2}$, include absolute value: $\sqrt{a^2} = |a|$.

Mathematical Notation

$\sqrt{x}$ means the square root of $x$$\times$ or $*$ represents multiplication$\div$ or $/$ represents division$\neq$ means not equal toRemember to use proper notation when solving problems

Why It Works

Using exponent rules, $\sqrt{ab} = (ab)^{1/2} = a^{1/2} b^{1/2} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{\frac{a}{b}} = (a/b)^{1/2} = a^{1/2} b^{-1/2} = \frac{\sqrt{a}}{\sqrt{b}}$. This direct application of exponents underpins the product and quotient properties for non-negative real numbers.

Remember

Always apply the product and quotient properties to simplify calculations involving square roots.

Quick Reference

Product Property:$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$Quotient Property:$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Understanding Properties of Square Roots

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Beginner Explanation

Simple explanation with $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

Simplify $\sqrt{16 \times 9}$ using the product property.

A$12$B$3 \times 4$C$\sqrt{36}$D$5 \times 5$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You have a rectangular garden with an area of $\frac{144}{4}$ square meters. Find the side length using the quotient property.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

If $\sqrt{x \times y} = 10$ and $\sqrt{x} = 2$, find $\sqrt{y}$.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

If $\sqrt{a^2 \times b^2} = 36$, and $a = 3$, find $b$.

A$12$B$4$C$6$D$9$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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