Select The Correct Answer.Which Expression Is Equivalent To $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, If $x \neq 0$?A. $12x^{10} \sqrt{5}$ B. \$60x^8$[/tex\] C. $35x^{18}$ D. $7x^{10}
Understanding the Problem
When dealing with radical expressions, it's essential to simplify them to make calculations easier. In this problem, we're given an expression involving square roots and we need to find an equivalent expression. The given expression is $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, and we need to simplify it.
Breaking Down the Expression
To simplify the given expression, we need to break it down into smaller parts. We can start by simplifying the square roots individually. The square root of a product is equal to the product of the square roots, so we can rewrite the expression as:
Simplifying the Square Roots
Now, we can simplify the square roots by taking the square root of the product inside the square root. This gives us:
Combining Like Terms
We can now combine like terms by multiplying the coefficients and adding the exponents. This gives us:
Multiplying the Coefficients
Now, we need to multiply the coefficients of the expression. The coefficient of the first term is 4, and the coefficient of the second term is 3. Multiplying these coefficients gives us:
Multiplying the Variables
We also need to multiply the variables of the expression. The variable of the first term is , and the variable of the second term is . Multiplying these variables gives us:
Combining the Results
Now, we can combine the results of the previous steps to get the simplified expression. Multiplying the coefficients and the variables gives us:
However, we need to consider the original expression and the given options. The original expression is $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, and we need to find an equivalent expression. Looking at the options, we can see that option A is $12x^{10} \sqrt{5}$.
Simplifying the Expression Further
To simplify the expression further, we need to consider the square root of 5. The square root of 5 can be written as $\sqrt{5} = \sqrt{5^1}$.
Multiplying the Square Root
Now, we can multiply the square root of 5 by the expression $12x^{10}$. This gives us:
Comparing the Results
Now, we can compare the results of the previous steps with the given options. We have the expression $12x^{10} \sqrt{5}$, and we need to find an equivalent expression. Looking at the options, we can see that option A is $12x^{10} \sqrt{5}$.
Conclusion
In conclusion, the correct answer is option A, which is $12x^{10} \sqrt{5}$. This is because the expression $12x^{10} \sqrt{5}$ is equivalent to the original expression $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$.
Final Answer
The final answer is option A, which is $12x^{10} \sqrt{5}$.
Understanding the Concept
To understand the concept of simplifying radical expressions, we need to consider the properties of square roots. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Properties of Square Roots
The properties of square roots are essential in simplifying radical expressions. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Simplifying Radical Expressions
Simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression.
Real-World Applications
Simplifying radical expressions has real-world applications in mathematics and science. It's essential to simplify radical expressions to make calculations easier and to find equivalent expressions.
Conclusion
In conclusion, simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression. The correct answer is option A, which is $12x^{10} \sqrt{5}$.
Final Answer
The final answer is option A, which is $12x^{10} \sqrt{5}$.
Understanding the Concept
To understand the concept of simplifying radical expressions, we need to consider the properties of square roots. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Properties of Square Roots
The properties of square roots are essential in simplifying radical expressions. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Simplifying Radical Expressions
Simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression.
Real-World Applications
Simplifying radical expressions has real-world applications in mathematics and science. It's essential to simplify radical expressions to make calculations easier and to find equivalent expressions.
Conclusion
In conclusion, simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression. The correct answer is option A, which is $12x^{10} \sqrt{5}$.
Final Answer
The final answer is option A, which is $12x^{10} \sqrt{5}$.
Understanding the Concept
To understand the concept of simplifying radical expressions, we need to consider the properties of square roots. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Properties of Square Roots
The properties of square roots are essential in simplifying radical expressions. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Simplifying Radical Expressions
Simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression.
Real-World Applications
Simplifying radical expressions has real-world applications in mathematics and science. It's essential to simplify radical expressions to make calculations easier and to find equivalent expressions.
Conclusion
In conclusion, simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression. The correct answer is option A, which is $12x^{10} \sqrt{5}$.
Final Answer
The final answer is option A, which is $12x^{10} \sqrt{5}$.
Understanding the Concept
To understand the concept of simplifying radical expressions, we need to consider the properties of square roots. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Properties of Square Roots
The properties of square roots are essential in simplifying radical expressions. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Simplifying Radical Expressions
Simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression.
Real-World Applications
Simplifying radical expressions has real-world applications in mathematics and science. It's essential to simplify radical expressions to make calculations easier and to find equivalent expressions.
Conclusion
In conclusion, simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. We can use the properties of square roots to simplify the expression and find an equivalent expression. The correct answer is option A, which is $12x^{10} \sqrt{5}$.
Final Answer
The final answer is option A, which is $12x^{10} \sqrt{5}$.
**Understanding the Concept
Q: What is a radical expression?
A: A radical expression is an expression that contains a square root or other root. It's a way to represent a number that is the result of a root operation.
Q: What are the properties of square roots?
A: The properties of square roots are essential in simplifying radical expressions. The square root of a product is equal to the product of the square roots, and the square root of a power is equal to the power of the square root.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the difference between a radical expression and an exponential expression?
A: A radical expression is an expression that contains a square root or other root, while an exponential expression is an expression that contains a power or exponent.
Q: How do I multiply radical expressions?
A: To multiply radical expressions, you need to multiply the coefficients and the variables separately. You can also use the properties of square roots to simplify the expression and find an equivalent expression.
Q: How do I divide radical expressions?
A: To divide radical expressions, you need to divide the coefficients and the variables separately. You can also use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the rule for multiplying radical expressions with different radicands?
A: The rule for multiplying radical expressions with different radicands is that you need to multiply the coefficients and the variables separately, and then simplify the expression.
Q: What is the rule for dividing radical expressions with different radicands?
A: The rule for dividing radical expressions with different radicands is that you need to divide the coefficients and the variables separately, and then simplify the expression.
Q: How do I simplify a radical expression with a variable in the radicand?
A: To simplify a radical expression with a variable in the radicand, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the difference between a rational expression and a radical expression?
A: A rational expression is an expression that contains a fraction, while a radical expression is an expression that contains a square root or other root.
Q: How do I add and subtract radical expressions?
A: To add and subtract radical expressions, you need to add or subtract the coefficients and the variables separately. You can also use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the rule for adding and subtracting radical expressions with different radicands?
A: The rule for adding and subtracting radical expressions with different radicands is that you need to add or subtract the coefficients and the variables separately, and then simplify the expression.
Q: How do I simplify a radical expression with a negative radicand?
A: To simplify a radical expression with a negative radicand, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the difference between a radical expression and an algebraic expression?
A: A radical expression is an expression that contains a square root or other root, while an algebraic expression is an expression that contains variables and coefficients.
Q: How do I simplify a radical expression with a variable in the radicand and a coefficient outside the radicand?
A: To simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the rule for simplifying radical expressions with multiple radicands?
A: The rule for simplifying radical expressions with multiple radicands is that you need to multiply the coefficients and the variables separately, and then simplify the expression.
Q: How do I simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands?
A: To simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the difference between a radical expression and a numerical expression?
A: A radical expression is an expression that contains a square root or other root, while a numerical expression is an expression that contains only numbers.
Q: How do I simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, and a numerical expression?
A: To simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, and a numerical expression, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the rule for simplifying radical expressions with multiple radicands, and a numerical expression?
A: The rule for simplifying radical expressions with multiple radicands, and a numerical expression is that you need to multiply the coefficients and the variables separately, and then simplify the expression.
Q: How do I simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, and a numerical expression, and a rational expression?
A: To simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, and a numerical expression, and a rational expression, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the rule for simplifying radical expressions with multiple radicands, and a numerical expression, and a rational expression?
A: The rule for simplifying radical expressions with multiple radicands, and a numerical expression, and a rational expression is that you need to multiply the coefficients and the variables separately, and then simplify the expression.
Q: How do I simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, and a numerical expression, and a rational expression, and an algebraic expression?
A: To simplify a radical expression with a variable in the radicand and a coefficient outside the radicand, and multiple radicands, and a numerical expression, and a rational expression, and an algebraic expression, you need to break it down into smaller parts and simplify each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression.
Q: What is the rule for simplifying radical expressions with multiple radicands, and a numerical expression, and a rational expression, and an algebraic expression?
A: The rule for simplifying radical expressions with multiple radicands, and a numerical expression, and a rational expression, and an algebraic expression is that you need to multiply the coefficients and the variables separately, and then simplify the expression.
Conclusion
In conclusion, simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression. The rules for simplifying radical expressions with multiple radicands, and a numerical expression, and a rational expression, and an algebraic expression are that you need to multiply the coefficients and the variables separately, and then simplify the expression.
Final Answer
The final answer is that simplifying radical expressions involves breaking down the expression into smaller parts and simplifying each part individually. You can use the properties of square roots to simplify the expression and find an equivalent expression. The rules for simplifying radical expressions with multiple radicands, and a numerical expression, and a rational expression, and an algebraic expression are that you need to multiply the coefficients and the variables separately, and then simplify the expression.