Which Expression Is Equivalent To $\left(5 X^3\right)(2 X)^3$?A. $10 X^6$B. $ 30 X 6 30 X^6 30 X 6 [/tex]C. $40 X^6$D. $1,000 X^{12}$

Understanding the Problem

When simplifying algebraic expressions, it's essential to apply the rules of exponents and multiplication. In this problem, we're given the expression $\left(5 x^3\right)(2 x)^3$ and asked to find an equivalent expression from the given options.

Applying the Rules of Exponents

To simplify the given expression, we need to apply the rules of exponents. The first rule states that when multiplying two powers with the same base, we add the exponents. In this case, we have $\left(5 x^3\right)(2 x)^3$, where the base is $x$ and the exponents are $3$ and $3$, respectively.

Using the Power of a Product Rule

The power of a product rule states that when raising a product to a power, we raise each factor to that power. In this case, we have $(2 x)^3$, which can be rewritten as $2^3 x^3$ using the power of a product rule.

Simplifying the Expression

Now that we've applied the rules of exponents, we can simplify the expression. We have $\left(5 x^3\right)(23 x^3)$, which can be rewritten as $5 \cdot 2^3 \cdot x^3 \cdot x^3$ using the associative property of multiplication.

Applying the Product of Powers Rule

The product of powers rule states that when multiplying two powers with the same base, we add the exponents. In this case, we have $x^3 \cdot x^3$, which can be rewritten as $x^{3+3}$ using the product of powers rule.

Simplifying the Expression Further

Now that we've applied the product of powers rule, we can simplify the expression further. We have $5 \cdot 2^3 \cdot x^{3+3}$, which can be rewritten as $5 \cdot 8 \cdot x^6$ using the definition of exponentiation.

Evaluating the Expression

Finally, we can evaluate the expression by multiplying the coefficients and adding the exponents. We have $5 \cdot 8 \cdot x^6$, which can be rewritten as $40 x^6$.

Conclusion

In conclusion, the expression $\left(5 x^3\right)(2 x)^3$ is equivalent to $40 x^6$. This can be verified by applying the rules of exponents and multiplication, as shown in the previous steps.

Answer

The correct answer is:

• C. $40 x^6$

Why is this the correct answer?

This is the correct answer because we applied the rules of exponents and multiplication to simplify the given expression. We used the power of a product rule, the product of powers rule, and the associative property of multiplication to rewrite the expression in a simpler form.

What are the key concepts in this problem?

The key concepts in this problem are:

• Rules of exponents: These rules govern how to simplify expressions with exponents.
• Power of a product rule: This rule states that when raising a product to a power, we raise each factor to that power.
• Product of powers rule: This rule states that when multiplying two powers with the same base, we add the exponents.
• Associative property of multiplication: This property states that the order in which we multiply numbers does not change the result.

How can I apply these concepts to other problems?

You can apply these concepts to other problems by:

• Using the power of a product rule: When raising a product to a power, raise each factor to that power.
• Using the product of powers rule: When multiplying two powers with the same base, add the exponents.
• Using the associative property of multiplication: When multiplying numbers, the order in which you multiply them does not change the result.

What are some common mistakes to avoid?

Some common mistakes to avoid when simplifying algebraic expressions include:

• Forgetting to apply the rules of exponents: Make sure to apply the rules of exponents when simplifying expressions with exponents.
• Not using the power of a product rule: When raising a product to a power, make sure to raise each factor to that power.
• Not using the product of powers rule: When multiplying two powers with the same base, make sure to add the exponents.

How can I practice simplifying algebraic expressions?

You can practice simplifying algebraic expressions by:

• Working through practice problems: Try simplifying different expressions using the rules of exponents and multiplication.
• Using online resources: There are many online resources available that can help you practice simplifying algebraic expressions.
• Seeking help from a teacher or tutor: If you're struggling to simplify algebraic expressions, consider seeking help from a teacher or tutor.
  

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern how to simplify expressions with exponents. The main rules are:

• Product of powers rule: When multiplying two powers with the same base, add the exponents.
• Power of a product rule: When raising a product to a power, raise each factor to that power.
• Power of a power rule: When raising a power to a power, multiply the exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents when multiplying two powers with the same base. For example, $x^3 \cdot x^4 = x^{3+4} = x^7$.

Q: How do I apply the power of a product rule?

A: To apply the power of a product rule, raise each factor to the power when raising a product to a power. For example, $(2x)^3 = 2^3 \cdot x^3 = 8x^3$.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, multiply the exponents when raising a power to a power. For example, $(x²⁾3 = x^{2 \cdot 3} = x^6$.

Q: What is the associative property of multiplication?

A: The associative property of multiplication states that the order in which we multiply numbers does not change the result. For example, $(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)$.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, apply the rules of exponents and the associative property of multiplication. For example, $\left(5x^3\right)(2x)3 = 5 \cdot 2^3 \cdot x^3 \cdot x^3 = 5 \cdot 8 \cdot x^6 = 40x^6$.

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 apply the rules of exponents: Make sure to apply the rules of exponents when simplifying expressions with exponents.
• Not using the power of a product rule: When raising a product to a power, make sure to raise each factor to that power.
• Not using the product of powers rule: When multiplying two powers with the same base, make sure to add the exponents.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

• Working through practice problems: Try simplifying different expressions using the rules of exponents and multiplication.
• Using online resources: There are many online resources available that can help you practice simplifying algebraic expressions.
• Seeking help from a teacher or tutor: If you're struggling to simplify algebraic expressions, consider seeking help from a teacher or tutor.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Science and engineering: Algebraic expressions are used to model and solve problems in science and engineering.
• Finance: Algebraic expressions are used to calculate interest rates and investments.
• Computer programming: Algebraic expressions are used to write algorithms and solve problems in computer programming.

Q: How can I use technology to simplify algebraic expressions?

A: You can use technology to simplify algebraic expressions by:

• Using a calculator: Many calculators have built-in functions for simplifying algebraic expressions.
• Using a computer algebra system: Computer algebra systems, such as Mathematica or Maple, can simplify algebraic expressions and solve equations.
• Using online resources: There are many online resources available that can help you simplify algebraic expressions, including online calculators and algebraic expression simplifiers.