Understanding Square Roots and Simplification

Oct 13, 2025 by leolavanderia.com.br 139 views

Introduction

Understanding Square Roots and Simplification When dealing with square roots, it's essential to simplify expressions to make them easier to work with. In this discussion, we will simplify the expression $${ \frac{\sqrt{12}}{\sqrt{18}} = }$$. This involves breaking down the square roots into their prime factors and simplifying the resulting expression.

Breaking Down Square Roots

Prime Factorization of 12 and 18 To simplify the expression, we need to break down the square roots into their prime factors. The prime factorization of 12 is $2^2 \cdot 3$ , and the prime factorization of 18 is $2 \cdot 3^2$ . We can rewrite the expression as $${ \frac{\sqrt{2^2 \cdot 3}}{\sqrt{2 \cdot 3^2}} = }$$.

Simplifying the Expression

Using the Properties of Square Roots We can simplify the expression by using the properties of square roots. When we divide two square roots, we can divide the numbers inside the square roots. So, we can rewrite the expression as $${ \frac{\sqrt{2^2 \cdot 3}}{\sqrt{2 \cdot 3^2}} = \frac{2\sqrt{3}}{\sqrt{2} \cdot 3} = }$$.

Further Simplification

Canceling Out Common Factors We can further simplify the expression by canceling out common factors. The expression can be rewritten as $${ \frac{2\sqrt{3}}{\sqrt{2} \cdot 3} = \frac{2\sqrt{3}}{3\sqrt{2}} = \frac{2}{3}\sqrt{\frac{3}{2}} = }$$.

Final Simplification

Simplifying the Fraction Under the Square Root We can simplify the fraction under the square root by dividing the numerator and denominator by their greatest common divisor. The expression can be rewritten as $${ \frac{2}{3}\sqrt{\frac{3}{2}} = \frac{2}{3}\sqrt{\frac{3}{2}} = \frac{2}{3}\sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{6} = }$$.

Conclusion

Simplifying the Expression In conclusion, we have simplified the expression $${ \frac{\sqrt{12}}{\sqrt{18}} = }$$ by breaking down the square roots into their prime factors and simplifying the resulting expression. The final simplified expression is $${ \frac{\sqrt{6}}{6} = }$$.

Frequently Asked Questions

Common Questions and Answers Q: What is the prime factorization of 12? A: The prime factorization of 12 is $2^2 \cdot 3$ .

Q: What is the prime factorization of 18? A: The prime factorization of 18 is $2 \cdot 3^2$ .

Q: How do we simplify the expression when dividing two square roots? A: We can simplify the expression by dividing the numbers inside the square roots.

Q: How do we cancel out common factors in the expression? A: We can cancel out common factors by dividing the numerator and denominator by their greatest common divisor.

Final Answer

The Final Simplified Expression The final simplified expression is $${ \frac{\sqrt{6}}{6} = }$$.

Introduction

Frequently Asked Questions

Common Questions and Answers

Q: What is the prime factorization of 12?

A: The prime factorization of 12 is $2^2 \cdot 3$

The prime factorization of 12 is $2^2 \cdot 3$ . This means that 12 can be broken down into two factors: 2 and 3. The exponent 2 indicates that 2 is squared, or multiplied by itself.

Q: What is the prime factorization of 18?

A: The prime factorization of 18 is $2 \cdot 3^2$

The prime factorization of 18 is $2 \cdot 3^2$ . This means that 18 can be broken down into two factors: 2 and 3. The exponent 2 indicates that 3 is squared, or multiplied by itself.

Q: How do we simplify the expression when dividing two square roots?

A: We can simplify the expression by dividing the numbers inside the square roots

When dividing two square roots, we can simplify the expression by dividing the numbers inside the square roots. This is because the square root of a number is the same as the number raised to the power of 1/2.

Q: How do we cancel out common factors in the expression?

A: We can cancel out common factors by dividing the numerator and denominator by their greatest common divisor

When simplifying an expression, we can cancel out common factors by dividing the numerator and denominator by their greatest common divisor. This is a way of simplifying the expression by removing any common factors.

Q: What is the final simplified expression?

A: The final simplified expression is $${ \frac{\sqrt{6}}{6} = }$$

The final simplified expression is $${ \frac{\sqrt{6}}{6} = }$$. This is the result of simplifying the original expression by breaking down the square roots into their prime factors and simplifying the resulting expression.

Additional Questions and Answers

More Common Questions and Answers

Q: What is the difference between a square root and a square?

A: A square root is the opposite of a square

A square root is the opposite of a square. While a square is a number raised to the power of 2, a square root is a number raised to the power of 1/2.

Q: How do we simplify a square root expression?

A: We can simplify a square root expression by breaking down the number inside the square root into its prime factors

We can simplify a square root expression by breaking down the number inside the square root into its prime factors. This involves finding the prime factorization of the number and then simplifying the resulting expression.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder

The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. This is an important concept in simplifying expressions and finding the greatest common divisor.