Simplify The Equation: 34 X M × 48 = 24 X 3 X \sqrt{34 X^m} \times \sqrt{48} = 24 X \sqrt{3 X} 34 X M ​ × 48 ​ = 24 X 3 X ​

Introduction

Mathematics is a vast and complex subject that involves solving equations, inequalities, and other mathematical expressions. One of the fundamental concepts in mathematics is the concept of square roots, which is used to find the value of a number that, when multiplied by itself, gives the original number. In this article, we will simplify the given equation: $\sqrt{34 x^m} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Understanding the Given Equation

The given equation involves square roots, which can be simplified using various mathematical techniques. The equation is: $\sqrt{34 x^m} \times \sqrt{48} = 24 x \sqrt{3 x}$. To simplify this equation, we need to understand the properties of square roots and how they can be manipulated.

Properties of Square Roots

Square roots have several properties that can be used to simplify mathematical expressions. Some of the key properties of square roots include:

• Multiplication Property: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
• Division Property: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
• Power Property: $(\sqrt{a})^n = a^{\frac{n}{2}}$

Simplifying the Given Equation

To simplify the given equation, we can use the multiplication property of square roots. The equation is: $\sqrt{34 x^m} \times \sqrt{48} = 24 x \sqrt{3 x}$. Using the multiplication property, we can rewrite the equation as: $\sqrt{34 x^m \times 48} = 24 x \sqrt{3 x}$.

Expanding the Expression Under the Square Root

The expression under the square root can be expanded as: $34 x^m \times 48 = 1632 x^m$. Therefore, the equation becomes: $\sqrt{1632 x^m} = 24 x \sqrt{3 x}$.

Simplifying the Expression Under the Square Root

The expression under the square root can be simplified as: $\sqrt{1632 x^m} = \sqrt{2^4 \times 3^2 \times 17 \times x^m}$. Therefore, the equation becomes: $\sqrt{2^4 \times 3^2 \times 17 \times x^m} = 24 x \sqrt{3 x}$.

Canceling Out Common Factors

The equation can be simplified further by canceling out common factors. The expression on the left-hand side can be rewritten as: $\sqrt{2^4 \times 3^2 \times 17 \times x^m} = 2^2 \times 3 \times \sqrt{17 \times x^m}$. Therefore, the equation becomes: $2^2 \times 3 \times \sqrt{17 \times x^m} = 24 x \sqrt{3 x}$.

Equating the Expressions

The expressions on both sides of the equation can be equated. Therefore, we have: $2^2 \times 3 \times \sqrt{17 \times x^m} = 24 x \sqrt{3 x}$.

Canceling Out Common Factors

The equation can be simplified further by canceling out common factors. The expression on the left-hand side can be rewritten as: $2^2 \times 3 = 12$. Therefore, the equation becomes: $12 \times \sqrt{17 \times x^m} = 24 x \sqrt{3 x}$.

Dividing Both Sides by 12

To simplify the equation further, we can divide both sides by 12. Therefore, we have: $\sqrt{17 \times x^m} = 2 x \sqrt{3 x}$.

Squaring Both Sides

To eliminate the square root, we can square both sides of the equation. Therefore, we have: $17 \times x^m = 4 x^2 \times 3 x$.

Simplifying the Expression

The expression on the right-hand side can be simplified as: $4 x^2 \times 3 x = 12 x^3$. Therefore, the equation becomes: $17 \times x^m = 12 x^3$.

Equating the Exponents

The exponents on both sides of the equation can be equated. Therefore, we have: $m = 3$.

Substituting the Value of m

The value of m can be substituted into the original equation. Therefore, we have: $\sqrt{34 x^3} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Simplifying the Expression Under the Square Root

The expression under the square root can be simplified as: $\sqrt{34 x^3} = \sqrt{2 \times 17 \times x^3}$. Therefore, the equation becomes: $\sqrt{2 \times 17 \times x^3} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Canceling Out Common Factors

The equation can be simplified further by canceling out common factors. The expression on the left-hand side can be rewritten as: $\sqrt{2 \times 17 \times x^3} = \sqrt{2 \times 17 \times x^2 \times x}$. Therefore, the equation becomes: $\sqrt{2 \times 17 \times x^2 \times x} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Simplifying the Expression Under the Square Root

The expression under the square root can be simplified as: $\sqrt{2 \times 17 \times x^2 \times x} = \sqrt{2 \times 17 \times x^2} \times \sqrt{x}$. Therefore, the equation becomes: $\sqrt{2 \times 17 \times x^2} \times \sqrt{x} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Canceling Out Common Factors

The equation can be simplified further by canceling out common factors. The expression on the left-hand side can be rewritten as: $\sqrt{2 \times 17 \times x^2} = \sqrt{2 \times 17 \times x} \times \sqrt{x}$. Therefore, the equation becomes: $\sqrt{2 \times 17 \times x} \times \sqrt{x} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Simplifying the Expression Under the Square Root

The expression under the square root can be simplified as: $\sqrt{2 \times 17 \times x} = \sqrt{2 \times 17} \times \sqrt{x}$. Therefore, the equation becomes: $\sqrt{2 \times 17} \times \sqrt{x} \times \sqrt{x} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Canceling Out Common Factors

The equation can be simplified further by canceling out common factors. The expression on the left-hand side can be rewritten as: $\sqrt{2 \times 17} \times \sqrt{x} \times \sqrt{x} = \sqrt{2 \times 17} \times x \sqrt{x}$. Therefore, the equation becomes: $\sqrt{2 \times 17} \times x \sqrt{x} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Simplifying the Expression Under the Square Root

The expression under the square root can be simplified as: $\sqrt{48} = \sqrt{2^4 \times 3} = 2^2 \times \sqrt{3}$. Therefore, the equation becomes: $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3} = 24 x \sqrt{3 x}$.

Canceling Out Common Factors

The equation can be simplified further by canceling out common factors. The expression on the left-hand side can be rewritten as: $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3} = \sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}$. Therefore, the equation becomes: $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3} = 24 x \sqrt{3 x}$.

Dividing Both Sides by $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}$

To simplify the equation further, we can divide both sides by $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}$. Therefore, we have: $1 = \frac{24 x \sqrt{3 x}}{\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}}$.

Simplifying the Expression

The expression on the right-hand side can be simplified as: $\frac{24 x \sqrt{3 x}}{\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2

Introduction

Mathematics is a vast and complex subject that involves solving equations, inequalities, and other mathematical expressions. One of the fundamental concepts in mathematics is the concept of square roots, which is used to find the value of a number that, when multiplied by itself, gives the original number. In this article, we will simplify the given equation: $\sqrt{34 x^m} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Q&A

Q: What is the first step in simplifying the given equation?

A: The first step in simplifying the given equation is to use the multiplication property of square roots, which states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.

Q: How do we simplify the expression under the square root?

A: We can simplify the expression under the square root by factoring out the common factors. In this case, we can rewrite the expression as: $\sqrt{34 x^m \times 48} = 24 x \sqrt{3 x}$.

Q: What is the next step in simplifying the equation?

A: The next step in simplifying the equation is to equate the exponents on both sides of the equation. Therefore, we have: $m = 3$.

Q: How do we substitute the value of m into the original equation?

A: We can substitute the value of m into the original equation by replacing $m$ with $3$. Therefore, we have: $\sqrt{34 x^3} \times \sqrt{48} = 24 x \sqrt{3 x}$.

Q: What is the final step in simplifying the equation?

A: The final step in simplifying the equation is to cancel out common factors and simplify the expression under the square root. Therefore, we have: $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3} = 24 x \sqrt{3 x}$.

Q: How do we solve for x?

A: To solve for x, we can divide both sides of the equation by $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}$. Therefore, we have: $1 = \frac{24 x \sqrt{3 x}}{\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}}$.

Q: What is the final solution to the equation?

A: The final solution to the equation is: $x = \frac{\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}}{24 \sqrt{3 x}}$.

Conclusion

Simplifying mathematical expressions is an essential skill in mathematics. By using the properties of square roots and simplifying the expression under the square root, we can solve complex equations and find the value of unknown variables. In this article, we simplified the given equation: $\sqrt{34 x^m} \times \sqrt{48} = 24 x \sqrt{3 x}$, and found the final solution to the equation.

Frequently Asked Questions

Q: What is the concept of square roots?

A: The concept of square roots is used to find the value of a number that, when multiplied by itself, gives the original number.

Q: What is the multiplication property of square roots?

A: The multiplication property of square roots states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.

Q: How do we simplify the expression under the square root?

A: We can simplify the expression under the square root by factoring out the common factors.

Q: What is the next step in simplifying the equation?

A: The next step in simplifying the equation is to equate the exponents on both sides of the equation.

Q: How do we substitute the value of m into the original equation?

A: We can substitute the value of m into the original equation by replacing $m$ with $3$.

Q: What is the final step in simplifying the equation?

A: The final step in simplifying the equation is to cancel out common factors and simplify the expression under the square root.

Q: How do we solve for x?

A: To solve for x, we can divide both sides of the equation by $\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}$.

Q: What is the final solution to the equation?

A: The final solution to the equation is: $x = \frac{\sqrt{2 \times 17} \times x \sqrt{x} \times 2^2 \times \sqrt{3}}{24 \sqrt{3 x}}$.

References

• [1] Khan Academy. (n.d.). Square Roots. Retrieved from https://www.khanacademy.org/math/algebra/x2f2f7d6/x2f2f7d7
• [2] Math Open Reference. (n.d.). Square Root. Retrieved from https://www.mathopenref.com/sqrt.html
• [3] Purplemath. (n.d.). Square Roots. Retrieved from https://www.purplemath.com/modules/sqrts.htm

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