Understanding the Concept of Product of Powers

Oct 13, 2025 by miodeya.com 136 views

Understanding the Concept of Product of Powers

When dealing with exponents, it's essential to understand the concept of product of powers. This concept allows us to simplify expressions by combining the exponents of the same base. In this article, we will explore how to simplify the product of powers $\left(3 n^4\right)\left(4 n^7\right)$ .

The Rule of Product of Powers

The rule of product of powers states that when multiplying two powers with the same base, we add the exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

where $a$ is the base and $m$ and $n$ are the exponents.

Applying the Rule to the Given Expression

Now, let's apply the rule of product of powers to the given expression $\left(3 n^4\right)\left(4 n^7\right)$ . We can start by identifying the bases and exponents in each term.

The first term is $3 n^4$ , where the base is $n$ and the exponent is $4$ .
The second term is $4 n^7$ , where the base is $n$ and the exponent is $7$ .

Simplifying the Expression

Using the rule of product of powers, we can simplify the expression by adding the exponents of the same base.

$\left(3 n^4\right)\left(4 n^7\right) = 3 \cdot 4 \cdot n^{4+7}$

Evaluating the Exponents

Now, let's evaluate the exponents by adding $4$ and $7$ .

$4+7 = 11$

Simplifying the Expression Further

Now that we have the evaluated exponent, we can simplify the expression further.

$\left(3 n^4\right)\left(4 n^7\right) = 3 \cdot 4 \cdot n^{11}$

Combining the Coefficients

Finally, we can combine the coefficients $3$ and $4$ by multiplying them.

$3 \cdot 4 = 12$

Simplifying the Expression to its Final Form

Now that we have combined the coefficients, we can simplify the expression to its final form.

$\left(3 n^4\right)\left(4 n^7\right) = 12 n^{11}$

Conclusion

In this article, we have explored how to simplify the product of powers $\left(3 n^4\right)\left(4 n^7\right)$ using the rule of product of powers. We have identified the bases and exponents in each term, applied the rule to simplify the expression, evaluated the exponents, and combined the coefficients to arrive at the final form of the expression.

Examples of Product of Powers

Here are some examples of product of powers:

$\left(2 x^3\right)\left(3 x^5\right) = 6 x^{3+5} = 6 x^8$
$\left(4 y^2\right)\left(3 y^4\right) = 12 y^{2+4} = 12 y^6$
$\left(2 z^3\right)\left(3 z^5\right) = 6 z^{3+5} = 6 z^8$

Tips for Simplifying Product of Powers

Here are some tips for simplifying product of powers:

Identify the bases and exponents in each term.
Apply the rule of product of powers by adding the exponents of the same base.
Evaluate the exponents by adding the numbers.
Combine the coefficients by multiplying them.
Simplify the expression to its final form.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying product of powers:

Failing to identify the bases and exponents in each term.
Applying the rule of product of powers incorrectly.
Failing to evaluate the exponents correctly.
Failing to combine the coefficients correctly.
Simplifying the expression incorrectly.

Conclusion

In conclusion, simplifying product of powers is an essential skill in algebra that can be used to simplify complex expressions. By following the rule of product of powers and applying it correctly, we can simplify expressions and arrive at the final form. Remember to identify the bases and exponents in each term, apply the rule, evaluate the exponents, combine the coefficients, and simplify the expression to its final form.

Frequently Asked Questions

In this article, we will answer some frequently asked questions about product of powers.

Q: What is the rule of product of powers?

A: The rule of product of powers states that when multiplying two powers with the same base, we add the exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

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

A: To apply the rule of product of powers, you need to identify the bases and exponents in each term. Then, add the exponents of the same base and simplify the expression.

Q: What if the bases are different?

A: If the bases are different, you cannot apply the rule of product of powers. In this case, you need to multiply the coefficients and add the exponents separately.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents by applying the rule of product of powers. For example:

$\left(2 x^{-3}\right)\left(3 x^{-2}\right) = 6 x^{-3-2} = 6 x^{-5}$

Q: How do I simplify an expression with fractional exponents?

A: To simplify an expression with fractional exponents, you need to apply the rule of product of powers and then simplify the resulting expression. For example:

$\left(2 x^{1/2}\right)\left(3 x^{1/3}\right) = 6 x^{1/2+1/3} = 6 x^{5/6}$

Q: Can I simplify an expression with multiple bases?

A: Yes, you can simplify an expression with multiple bases by applying the rule of product of powers separately for each base. For example:

$\left(2 x^3\right)\left(3 y^2\right)\left(4 z^5\right) = 24 x^3 y^2 z^5$

Q: How do I simplify an expression with variables in the exponents?

A: To simplify an expression with variables in the exponents, you need to apply the rule of product of powers and then simplify the resulting expression. For example:

$\left(2 x^a\right)\left(3 x^b\right) = 6 x^{a+b}$

Q: Can I simplify an expression with complex numbers in the exponents?

A: Yes, you can simplify an expression with complex numbers in the exponents by applying the rule of product of powers and then simplifying the resulting expression. For example:

$\left(2 x^{3+4i}\right)\left(3 x^{2-5i}\right) = 6 x^{5+9i}$

Q: How do I simplify an expression with multiple terms with the same base?

A: To simplify an expression with multiple terms with the same base, you need to apply the rule of product of powers separately for each term and then combine the results. For example:

$\left(2 x^3\right)\left(3 x^3\right)\left(4 x^3\right) = 24 x^{3+3+3} = 24 x^9$

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by applying the rule of product of powers. For example:

$\left(2 x^0\right)\left(3 x^0\right) = 6 x^{0+0} = 6 x^0 = 6$

Q: How do I simplify an expression with a negative exponent and a variable in the exponent?

A: To simplify an expression with a negative exponent and a variable in the exponent, you need to apply the rule of product of powers and then simplify the resulting expression. For example:

$\left(2 x^{-a}\right)\left(3 x^b\right) = 6 x^{-a+b}$

Q: Can I simplify an expression with a complex number in the exponent and a variable in the exponent?

A: Yes, you can simplify an expression with a complex number in the exponent and a variable in the exponent by applying the rule of product of powers and then simplifying the resulting expression. For example:

$\left(2 x^{3+4i-a}\right)\left(3 x^{2-5i+b}\right) = 6 x^{5+9i-a+b}$

Conclusion

In this article, we have answered some frequently asked questions about product of powers. We have covered topics such as the rule of product of powers, applying the rule, simplifying expressions with negative exponents, fractional exponents, multiple bases, variables in the exponents, complex numbers in the exponents, and multiple terms with the same base. We hope that this article has been helpful in clarifying any doubts you may have had about product of powers.