Select The Correct Answer.Simplify The Following Algebraic Expression: $\sqrt{180 X^{25} Y^{23} Z^{39}}$A. $36 X^{12} Y^{11} Z^{19} \sqrt{5 X Y Z}$ B. $x^{12} Y^{11} Z^{19} \sqrt{36 X Y Z}$ C. $6 X^{12} Y^{11} Z^{19}

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given algebraic expression: $\sqrt{180 x^{25} y^{23} z^{39}}$. We will break down the expression into smaller parts, apply the necessary rules, and arrive at the simplified form.

Understanding the Expression

The given expression is $\sqrt{180 x^{25} y^{23} z^{39}}$. To simplify this expression, we need to understand the properties of radicals and exponents. The expression inside the square root can be broken down into its prime factors.

Breaking Down the Expression

Let's start by breaking down the expression inside the square root:

$\sqrt{180 x^{25} y^{23} z^{39}}$

We can rewrite 180 as $2^2 \cdot 3^2 \cdot 5$. Therefore, the expression becomes:

$\sqrt{(2^2 \cdot 3^2 \cdot 5) x^{25} y^{23} z^{39}}$

Now, we can separate the expression into two parts: the part inside the square root and the part outside the square root.

Separating the Expression

The part inside the square root is:

$(2^2 \cdot 3^2 \cdot 5) x^{25} y^{23} z^{39}$

The part outside the square root is:

$\sqrt{x^{25} y^{23} z^{39}}$

Applying the Rules of Exponents

Now, let's apply the rules of exponents to simplify the expression inside the square root:

$(2^2 \cdot 3^2 \cdot 5) x^{25} y^{23} z^{39}$

We can rewrite this expression as:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39}$

Using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39} = 2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39}$

Now, let's focus on the part outside the square root:

$\sqrt{x^{25} y^{23} z^{39}}$

We can rewrite this expression as:

$x^{25/2} \cdot y^{23/2} \cdot z^{39/2}$

Simplifying the Expression

Now, let's simplify the expression by combining the two parts:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39} \cdot x^{25/2} \cdot y^{23/2} \cdot z^{39/2}$

Using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39} \cdot x^{25/2} \cdot y^{23/2} \cdot z^{39/2} = 2^2 \cdot 3^2 \cdot 5 \cdot x^{25+25/2} \cdot y^{23+23/2} \cdot z^{39+39/2}$

Simplifying the exponents, we get:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{50/2} \cdot y^{46/2} \cdot z^{78/2}$

Further simplifying, we get:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39}$

Now, let's simplify the expression by combining the two parts:

$2^2 \cdot 3^2 \cdot 5 \cdot x^{25} \cdot y^{23} \cdot z^{39} = 36 \cdot x^{25} \cdot y^{23} \cdot z^{39}$

The Final Answer

The simplified expression is:

$36 x^{12} y^{11} z^{19} \sqrt{5 x y z}$

This is the correct answer.

Conclusion

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Introduction

In our previous article, we simplified the given algebraic expression: $\sqrt{180 x^{25} y^{23} z^{39}}$. We broke down the expression into smaller parts, applied the necessary rules, and arrived at the simplified form. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to break down the expression into smaller parts. This involves identifying the prime factors of the numbers and the exponents of the variables.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to apply the rules of exponents. This involves combining the exponents of the variables using the rule $a^m \cdot a^n = a^{m+n}$.

Q: What is the difference between a radical and an exponent?

A: A radical is a symbol that represents the square root of a number, while an exponent is a number that represents the power to which a number is raised.

Q: How do I simplify an expression with a radical in the denominator?

A: To simplify an expression with a radical in the denominator, you need to rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression is obtained by changing the sign of the second term. For example, the conjugate of $a + b$ is $a - b$.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to apply the rules of radicals. This involves combining the radicals using the rule $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Q: What is the rule for multiplying radicals?

A: The rule for multiplying radicals is $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Q: How do I simplify an expression with a radical in the numerator and a rational number in the denominator?

A: To simplify an expression with a radical in the numerator and a rational number in the denominator, you need to rationalize the numerator. This involves multiplying the numerator and denominator by the conjugate of the numerator.

Q: What is the difference between rationalizing the numerator and rationalizing the denominator?

A: Rationalizing the numerator involves multiplying the numerator and denominator by the conjugate of the numerator, while rationalizing the denominator involves multiplying the numerator and denominator by the conjugate of the denominator.

Q: How do I simplify an expression with multiple rational numbers?

A: To simplify an expression with multiple rational numbers, you need to apply the rules of rational numbers. This involves combining the rational numbers using the rule $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.

Q: What is the rule for dividing rational numbers?

A: The rule for dividing rational numbers is $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By breaking down the expression into smaller parts, applying the necessary rules, and combining the two parts, we can arrive at the simplified form. In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We hope that this article has been helpful in clarifying any doubts you may have had.