Type The Correct Answer In The Box. Use Numerals Instead Of Words.Consider This Expression: \[$\sqrt{x^4} - Y^2\$\].When \[$x = 3\$\] And \[$y = -6\$\], The Value Of The Expression Is \[$\square\$\].
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Understanding the Expression
When dealing with algebraic expressions, it's essential to simplify them to make calculations easier. The given expression is {\sqrt{x^4} - y^2$}$. To simplify this expression, we need to understand the properties of exponents and square roots.
Properties of Exponents
The expression {\sqrt{x^4}$}$ can be simplified using the property of exponents that states {\sqrt{a^n} = a^{n/2}$}$. Applying this property, we get {\sqrt{x^4} = x^{4/2} = x^2$}$.
Simplifying the Expression
Now that we have simplified the square root term, we can rewrite the original expression as {x^2 - y^2$}$. This expression can be further simplified using the difference of squares formula, which states that {a^2 - b^2 = (a + b)(a - b)$}$.
Applying the Difference of Squares Formula
Applying the difference of squares formula to the expression {x^2 - y^2$}$, we get {(x + y)(x - y)$}$.
Substituting the Given Values
Now that we have simplified the expression, we can substitute the given values of {x = 3$}$ and {y = -6$}$ to find the value of the expression.
Calculating the Value of the Expression
Substituting the given values, we get {(3 + (-6))(3 - (-6)) = (-3)(9) = -27$}$.
Conclusion
In conclusion, simplifying algebraic expressions is an essential skill in mathematics. By understanding the properties of exponents and square roots, we can simplify complex expressions and make calculations easier. In this example, we simplified the expression {\sqrt{x^4} - y^2$}$ and substituted the given values to find the value of the expression.
Final Answer
The final answer is: -27
Discussion
This problem requires the application of algebraic properties and formulas to simplify the given expression. The difference of squares formula is a useful tool for simplifying expressions of the form {a^2 - b^2$}$. By understanding and applying these formulas, we can simplify complex expressions and make calculations easier.
Related Problems
- Simplify the expression {\sqrt{x^2} - y^2$}$ using the properties of exponents and square roots.
- Apply the difference of squares formula to the expression {x^2 - y^2$}$ and simplify it.
- Substitute the given values of {x = 4$}$ and {y = 3$}$ into the expression {x^2 - y^2$}$ and calculate the value of the expression.
Practice Problems
- Simplify the expression {\sqrt{x^3} - y^3$}$ using the properties of exponents and square roots.
- Apply the difference of squares formula to the expression {x^2 - y^2$}$ and simplify it.
- Substitute the given values of {x = 2$}$ and {y = 5$}$ into the expression {x^2 - y^2$}$ and calculate the value of the expression.
Additional Resources
- Khan Academy: Algebraic Expressions
- Mathway: Simplifying Algebraic Expressions
- Wolfram Alpha: Algebraic Expressions
Glossary
- Algebraic Expression: An expression that contains variables and constants, and is used to represent a value or a relationship between values.
- Exponent: A small number that is raised to a power, indicating how many times the base number should be multiplied by itself.
- Square Root: A number that, when multiplied by itself, gives the original number.
- Difference of Squares Formula: A formula that states {a^2 - b^2 = (a + b)(a - b)$}$, which can be used to simplify expressions of the form {a^2 - b^2$}$.
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Q: What is an algebraic expression?
A: An algebraic expression is a mathematical expression that contains variables and constants, and is used to represent a value or a relationship between values.
Q: What are the properties of exponents?
A: The properties of exponents state that:
- {\sqrt{a^n} = a^{n/2}$]
- [$a^m \cdot a^n = a^{m+n}$]
- [$a^m / a^n = a^{m-n}$]
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you can use the following steps:
- Identify any like terms in the expression.
- Combine the like terms by adding or subtracting their coefficients.
- Simplify any expressions inside parentheses.
- Apply any exponent rules or properties.
- Simplify the expression as much as possible.
Q: What is the difference of squares formula?
A: The difference of squares formula states that:
[$a^2 - b^2 = (a + b)(a - b)$]
This formula can be used to simplify expressions of the form [a^2 - b^2\$}.
Q: How do I apply the difference of squares formula?
A: To apply the difference of squares formula, you can follow these steps:
- Identify the expression as a difference of squares.
- Factor the expression into the form [$(a + b)(a - b)$].
- Simplify the expression as much as possible.
Q: What are some common algebraic expressions?
A: Some common algebraic expressions include:
- [$x^2 + 4x + 4$]
- [$x^2 - 4x + 4$]
- [$x^2 + 2x - 3$]
- [$x^2 - 2x - 3$]
Q: How do I simplify a square root expression?
A: To simplify a square root expression, you can use the following steps:
- Identify any perfect squares inside the square root.
- Simplify the expression by taking the square root of the perfect square.
- Simplify the expression as much as possible.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Forgetting to combine like terms.
- Not simplifying expressions inside parentheses.
- Not applying exponent rules or properties.
- Not simplifying the expression as much as possible.
Q: How do I check my work when simplifying algebraic expressions?
A: To check your work when simplifying algebraic expressions, you can follow these steps:
- Plug in some values for the variables.
- Simplify the expression using the given values.
- Check that the simplified expression is correct.
Q: What are some real-world applications of simplifying algebraic expressions?
A: Simplifying algebraic expressions has many real-world applications, including:
- Solving systems of equations.
- Finding the maximum or minimum value of a function.
- Modeling real-world phenomena, such as population growth or chemical reactions.
Q: How do I practice simplifying algebraic expressions?
A: To practice simplifying algebraic expressions, you can try the following:
- Work through practice problems in a textbook or online resource.
- Use online tools or software to generate random algebraic expressions.
- Ask a teacher or tutor for help or feedback.
Q: What are some additional resources for learning about simplifying algebraic expressions?
A: Some additional resources for learning about simplifying algebraic expressions include:
- Khan Academy: Algebraic Expressions
- Mathway: Simplifying Algebraic Expressions
- Wolfram Alpha: Algebraic Expressions
Q: How do I know if I have simplified an algebraic expression correctly?
A: To know if you have simplified an algebraic expression correctly, you can follow these steps:
- Check that you have combined like terms correctly.
- Check that you have simplified expressions inside parentheses correctly.
- Check that you have applied exponent rules or properties correctly.
- Check that you have simplified the expression as much as possible.
Q: What are some common algebraic expressions that can be simplified using the difference of squares formula?
A: Some common algebraic expressions that can be simplified using the difference of squares formula include:
- [$x^2 - 4$]
- [$x^2 - 9$]
- [$x^2 - 16$]
Q: How do I simplify an algebraic expression that contains a square root?
A: To simplify an algebraic expression that contains a square root, you can use the following steps:
- Identify any perfect squares inside the square root.
- Simplify the expression by taking the square root of the perfect square.
- Simplify the expression as much as possible.
Q: What are some common mistakes to avoid when simplifying algebraic expressions that contain square roots?
A: Some common mistakes to avoid when simplifying algebraic expressions that contain square roots include:
- Forgetting to identify perfect squares inside the square root.
- Not simplifying the expression by taking the square root of the perfect square.
- Not simplifying the expression as much as possible.
Q: How do I check my work when simplifying algebraic expressions that contain square roots?
A: To check your work when simplifying algebraic expressions that contain square roots, you can follow these steps:
- Plug in some values for the variables.
- Simplify the expression using the given values.
- Check that the simplified expression is correct.
Q: What are some real-world applications of simplifying algebraic expressions that contain square roots?
A: Simplifying algebraic expressions that contain square roots has many real-world applications, including:
- Solving systems of equations that contain square roots.
- Finding the maximum or minimum value of a function that contains square roots.
- Modeling real-world phenomena, such as population growth or chemical reactions, that contain square roots.