Simplify 169 A 4 B C 3 \sqrt{169 A^4 B C^3} 169 A 4 B C 3 .A) 13 A 4 C 2 B C 13 A^4 C^2 \sqrt{b C} 13 A 4 C 2 B C B) 13 A C A 2 B C 13 A C \sqrt{a^2 B C} 13 A C A 2 B C C) 13 A 2 C B C 13 A^2 C \sqrt{b C} 13 A 2 C B C D) 13 A A 2 B C 3 13 A \sqrt{a^2 B C^3} 13 A A 2 B C 3

Mar 13, 2025 by ADMIN 283 views

$Simplify $\sqrt{169 a^4 b c^3}$$

Understanding the Problem

When simplifying a square root expression, we need to identify perfect squares within the expression and take their square roots. This process involves factoring the expression into its prime factors and then grouping the factors in pairs of identical factors. The square root of a product is the product of the square roots, and the square root of a power is the power of the square root.

Breaking Down the Expression

To simplify $\sqrt{169 a^4 b c^3}$ , we need to break down the expression into its prime factors. We can start by factoring the number 169, which is a perfect square. The prime factorization of 169 is $13^2$ . We can also factor the variables $a^4$ , $b$ , and $c^3$ .

Factoring the Expression

The expression $\sqrt{169 a^4 b c^3}$ can be factored as follows:

\sqrt{169 a^4 b c^3} = \sqrt{(13^2) (a^4) (b) (c^3)}

Grouping Factors

We can group the factors in pairs of identical factors:

\sqrt{(13^2) (a^4) (b) (c^3)} = \sqrt{(13^2) (a^4) (b) (c^2) (c)}

Taking Square Roots

Now we can take the square roots of the perfect squares:

\sqrt{(13^2) (a^4) (b) (c^2) (c)} = 13 a^2 c \sqrt{b c}

Comparing with the Options

Comparing our simplified expression with the options, we can see that the correct answer is:

C) $13 a^2 c \sqrt{b c}$

Conclusion

Simplifying a square root expression involves identifying perfect squares within the expression and taking their square roots. By factoring the expression into its prime factors and grouping the factors in pairs of identical factors, we can simplify the expression and find the correct answer.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

Factor the number 169 into its prime factors: $13^2$ .
Factor the variables $a^4$ , $b$ , and $c^3$ .
Group the factors in pairs of identical factors: $(13^2) (a^4) (b) (c^2) (c)$ .
Take the square roots of the perfect squares: $13 a^2 c \sqrt{b c}$ .

Common Mistakes

When simplifying a square root expression, it's easy to make mistakes. Here are some common mistakes to avoid:

Not factoring the expression into its prime factors.
Not grouping the factors in pairs of identical factors.
Not taking the square roots of the perfect squares.

Real-World Applications

Simplifying a square root expression has many real-world applications. For example, in physics, we often need to simplify expressions involving square roots to solve problems involving motion and energy. In engineering, we need to simplify expressions involving square roots to design and build structures and machines.

Practice Problems

Here are some practice problems to help you practice simplifying square root expressions:

Simplify $\sqrt{25 x^4 y^2 z^3}$ .
Simplify $\sqrt{36 a^3 b^2 c^4}$ .
Simplify $\sqrt{49 d^2 e^4 f^3}$ .

Conclusion

Frequently Asked Questions

Q: What is the first step in simplifying a square root expression?

A: The first step in simplifying a square root expression is to factor the expression into its prime factors.

Q: How do I factor the expression into its prime factors?

A: To factor the expression into its prime factors, you need to identify the prime factors of each number and variable in the expression. For example, the prime factorization of 169 is $13^2$ , and the prime factorization of $a^4$ is $a^4$ .

Q: What is the next step after factoring the expression into its prime factors?

A: After factoring the expression into its prime factors, you need to group the factors in pairs of identical factors. This will help you identify the perfect squares within the expression.

Q: How do I group the factors in pairs of identical factors?

A: To group the factors in pairs of identical factors, you need to look for pairs of identical factors in the expression. For example, if you have the expression $(13^2) (a^4) (b) (c^2) (c)$ , you can group the factors as follows: $(13^2) (a^4) (b) (c^2) (c) = (13^2) (a^4) (b) (c^2) (c)$ .

Q: What is the final step in simplifying a square root expression?

A: The final step in simplifying a square root expression is to take the square roots of the perfect squares. This will give you the simplified expression.

Q: How do I take the square roots of the perfect squares?

A: To take the square roots of the perfect squares, you need to look for the perfect squares within the expression and take their square roots. For example, if you have the expression $(13^2) (a^4) (b) (c^2) (c)$ , you can take the square roots of the perfect squares as follows: $\sqrt{(13^2) (a^4) (b) (c^2) (c)} = 13 a^2 c \sqrt{b c}$ .

Q: What are some common mistakes to avoid when simplifying a square root expression?

A: Some common mistakes to avoid when simplifying a square root expression include not factoring the expression into its prime factors, not grouping the factors in pairs of identical factors, and not taking the square roots of the perfect squares.

Q: What are some real-world applications of simplifying a square root expression?

A: Simplifying a square root expression has many real-world applications, including physics and engineering. In physics, we often need to simplify expressions involving square roots to solve problems involving motion and energy. In engineering, we need to simplify expressions involving square roots to design and build structures and machines.

Q: How can I practice simplifying a square root expression?

A: You can practice simplifying a square root expression by working through practice problems, such as simplifying $\sqrt{25 x^4 y^2 z^3}$ or $\sqrt{36 a^3 b^2 c^4}$ .

Additional Resources

For more information on simplifying a square root expression, see the article "Simplify $\sqrt{169 a^4 b c^3}$ ".
For practice problems, see the article "Practice Problems: Simplifying a Square Root Expression".
For more information on real-world applications of simplifying a square root expression, see the article "Real-World Applications of Simplifying a Square Root Expression".