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Surd Multiplication Law Calculator

Rule for multiplying square roots.

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Resulting Value

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Overview

The Surd Multiplication Law states that the product of the square roots of two non-negative numbers is equal to the square root of their product. This identity is a specific application of the power of a product rule in exponentiation, where the radical is treated as an exponent of one-half.

Symbols

Variables

a = First Value, b = Second Value, = Resulting Value

First Value
Variable
Second Value
Variable
Resulting Value
Variable

Apply it well

When To Use

When to use: This law is applied when simplifying products of radicals or when consolidating multiple surds into a single term for easier calculation. It is strictly applicable to non-negative real numbers to avoid encountering complex numbers in basic real-number arithmetic.

Why it matters: This principle is vital for maintaining exact values in geometry and trigonometry, such as calculating side lengths from areas without rounding decimals. It also facilitates the rationalization of denominators and the simplification of quadratic solutions, ensuring mathematical expressions are standardized.

Avoid these traps

Common Mistakes

  • Adding numbers inside square roots: sqrt(a) + sqrt(b) != sqrt(a+b).
  • Convert units and scales before substituting, especially percentages, time units, or powers of ten.
  • Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

One free problem

Practice Problem

Practice Problem 1

Calculate the result of multiplying the square root of 2 by the square root of 18.

First Value2
Second Value18

Solve for: result

Hint: Combine the numbers under a single square root sign first.

Practice Problem 2

If the product of √a and √5 is equal to 10, find the value of the variable a.

Second Value5
Resulting Value10

Solve for:

Hint: Square both sides of the consolidated equation to eliminate the radical.

Practice Problem 3

Determine the value of b given that √27 ×√b = 9.

First Value27
Resulting Value9

Solve for:

Hint: Note that 9 is the square root of 81.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Square root
  2. Britannica: Square root
  3. Halliday, Resnick, Walker, Fundamentals of Physics
  4. Bird, Stewart, Lightfoot, Transport Phenomena
  5. Stewart, James. Calculus: Early Transcendentals
  6. Stewart, James. Calculus: Early Transcendentals. Cengage Learning.
  7. AQA GCSE Maths — Number (Surds)