Which Equation Is Equivalent To $P=210 X^{\frac{4}{3}} Y^{\frac{7}{3}}$?1. $P=\sqrt[3]{210 X^4 Y^7}$2. $ P = 70 X Y 2 X Y 3 P=70 X Y^2 \sqrt[3]{x Y} P = 70 X Y 2 3 X Y ​ [/tex]3. $P=210 X Y^2 \sqrt[3]{x Y}$4. $P=210 X Y^2 \sqrt[3]{x^3

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Introduction

Exponential equations are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will explore the process of simplifying exponential equations, focusing on the given equation $P=210 x^{\frac{4}{3}} y^{\frac{7}{3}}$ and determining which of the provided options is equivalent to it.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and they can be simplified using various techniques. The given equation $P=210 x^{\frac{4}{3}} y^{\frac{7}{3}}$ is an example of an exponential equation, where the variable P is a function of x and y raised to certain powers.

Simplifying the Given Equation

To simplify the given equation, we can use the properties of exponents. Specifically, we can use the rule that states $a^m \cdot a^n = a^{m+n}$, where a is a non-zero number and m and n are integers.

Using this rule, we can rewrite the given equation as follows:

P=210x43y73=210(x43⋅y73)P=210 x^{\frac{4}{3}} y^{\frac{7}{3}} = 210 (x^{\frac{4}{3}} \cdot y^{\frac{7}{3}})

Now, we can simplify the expression inside the parentheses using the rule mentioned above:

x43⋅y73=(x13⋅y13)4⋅(x13⋅y13)3x^{\frac{4}{3}} \cdot y^{\frac{7}{3}} = (x^{\frac{1}{3}} \cdot y^{\frac{1}{3}})^4 \cdot (x^{\frac{1}{3}} \cdot y^{\frac{1}{3}})^3

Simplifying further, we get:

x43⋅y73=(x13⋅y13)7x^{\frac{4}{3}} \cdot y^{\frac{7}{3}} = (x^{\frac{1}{3}} \cdot y^{\frac{1}{3}})^7

Now, we can rewrite the original equation as:

P=210(x13⋅y13)7P=210 (x^{\frac{1}{3}} \cdot y^{\frac{1}{3}})^7

Evaluating the Options

Now that we have simplified the given equation, we can evaluate the options provided to determine which one is equivalent to it.

Option 1: $P=\sqrt[3]{210 x^4 y^7}$

To evaluate this option, we can start by simplifying the expression inside the cube root:

210x4y73=(210x4y7)13\sqrt[3]{210 x^4 y^7} = (210 x^4 y^7)^{\frac{1}{3}}

Using the rule that states $(am)n = a^{m \cdot n}$, we can rewrite the expression as:

(210x4y7)13=21013x43y73(210 x^4 y^7)^{\frac{1}{3}} = 210^{\frac{1}{3}} x^{\frac{4}{3}} y^{\frac{7}{3}}

This expression is not equivalent to the simplified equation we obtained earlier.

Option 2: $P=70 x y^2 \sqrt[3]{x y}$

To evaluate this option, we can start by simplifying the expression inside the cube root:

xy3=(xy)13\sqrt[3]{x y} = (x y)^{\frac{1}{3}}

Using the rule that states $(am)n = a^{m \cdot n}$, we can rewrite the expression as:

(xy)13=x13y13(x y)^{\frac{1}{3}} = x^{\frac{1}{3}} y^{\frac{1}{3}}

Now, we can rewrite the original equation as:

P=70xy2(x13y13)P=70 x y^2 (x^{\frac{1}{3}} y^{\frac{1}{3}})

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

P=70xy2(x13y13)=70x43y73P=70 x y^2 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 70 x^{\frac{4}{3}} y^{\frac{7}{3}}

This expression is not equivalent to the simplified equation we obtained earlier.

Option 3: $P=210 x y^2 \sqrt[3]{x y}$

To evaluate this option, we can start by simplifying the expression inside the cube root:

xy3=(xy)13\sqrt[3]{x y} = (x y)^{\frac{1}{3}}

Using the rule that states $(am)n = a^{m \cdot n}$, we can rewrite the expression as:

(xy)13=x13y13(x y)^{\frac{1}{3}} = x^{\frac{1}{3}} y^{\frac{1}{3}}

Now, we can rewrite the original equation as:

P=210xy2(x13y13)P=210 x y^2 (x^{\frac{1}{3}} y^{\frac{1}{3}})

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

P=210xy2(x13y13)=210x43y73P=210 x y^2 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 210 x^{\frac{4}{3}} y^{\frac{7}{3}}

This expression is equivalent to the simplified equation we obtained earlier.

Option 4: $P=210 x y^2 \sqrt[3]{x^3}$

To evaluate this option, we can start by simplifying the expression inside the cube root:

x33=(x3)13\sqrt[3]{x^3} = (x^3)^{\frac{1}{3}}

Using the rule that states $(am)n = a^{m \cdot n}$, we can rewrite the expression as:

(x3)13=x1=x(x^3)^{\frac{1}{3}} = x^1 = x

Now, we can rewrite the original equation as:

P=210xy2xP=210 x y^2 x

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

P=210xy2x=210x2y2P=210 x y^2 x = 210 x^2 y^2

This expression is not equivalent to the simplified equation we obtained earlier.

Conclusion

In conclusion, the correct answer is Option 3: $P=210 x y^2 \sqrt[3]{x y}$, which is equivalent to the simplified equation we obtained earlier. This option correctly simplifies the expression inside the cube root and uses the rule that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression.

Final Answer

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable raised to a power. It is a mathematical expression that can be simplified using various techniques.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents, such as the rule that states $a^m \cdot a^n = a^{m+n}$, where a is a non-zero number and m and n are integers.

Q: What is the rule for simplifying exponential expressions inside parentheses?

A: The rule for simplifying exponential expressions inside parentheses is that $(am)n = a^{m \cdot n}$, where a is a non-zero number and m and n are integers.

Q: How do I simplify an expression with a cube root?

A: To simplify an expression with a cube root, you can rewrite it as $(am){\frac{1}{n}} = a^{\frac{m}{n}}$, where a is a non-zero number and m and n are integers.

Q: What is the difference between an exponential equation and a polynomial equation?

A: An exponential equation is an equation that involves a variable raised to a power, while a polynomial equation is an equation that involves a variable raised to various powers, with coefficients in front of each term.

Q: Can I use the same techniques to simplify polynomial equations as I do for exponential equations?

A: No, the techniques used to simplify exponential equations are not the same as those used to simplify polynomial equations. Polynomial equations require different techniques, such as factoring and using the distributive property.

Q: How do I determine which option is equivalent to a simplified exponential equation?

A: To determine which option is equivalent to a simplified exponential equation, you can start by simplifying the expression inside the parentheses or the cube root, and then use the properties of exponents to simplify the expression.

Q: What are some common mistakes to avoid when simplifying exponential equations?

A: Some common mistakes to avoid when simplifying exponential equations include:

  • Not using the correct properties of exponents
  • Not simplifying the expression inside the parentheses or the cube root
  • Not using the distributive property correctly
  • Not checking the options carefully before selecting the correct one

Q: How can I practice simplifying exponential equations?

A: You can practice simplifying exponential equations by working through examples and exercises, such as those found in a mathematics textbook or online resource. You can also try simplifying exponential equations on your own, using the techniques and properties of exponents that you have learned.

Q: What are some real-world applications of simplifying exponential equations?

A: Simplifying exponential equations has many real-world applications, such as:

  • Modeling population growth and decay
  • Calculating compound interest and investment returns
  • Analyzing data and making predictions
  • Solving problems in physics, engineering, and other fields

Conclusion

In conclusion, simplifying exponential equations is an important skill that has many real-world applications. By understanding the properties of exponents and using the correct techniques, you can simplify exponential equations and solve problems in a variety of fields.