Simplify The Expression. Express Your Answer Using Exponents.$\left(2 B^9 C^9 D^5\right)^3$\square$

Feb 28, 2025 by ADMIN 101 views

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Understanding Exponents and Their Application

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In the given expression, $\left(2 b^9 c^9 d^5\right)^3$ , we are required to simplify it using exponents. This involves applying the rules of exponents to rewrite the expression in a more compact and simplified form.

The Power of a Product Rule

The power of a product rule states that when a product of numbers is raised to a power, each number in the product is raised to that power. In other words, for any numbers $a$ and $b$ and any positive integer $n$ , we have:

\left(ab\right)^n = a^n b^n

Applying the Power of a Product Rule

Using the power of a product rule, we can rewrite the given expression as:

\left(2 b^9 c^9 d^5\right)^3 = 2^3 \left(b^9\right)^3 \left(c^9\right)^3 \left(d^5\right)^3

Simplifying the Expression

Now, we can simplify each term in the expression using the power of a product rule:

2^3 = 8

\left(b^9\right)^3 = b^{9 \cdot 3} = b^{27}

\left(c^9\right)^3 = c^{9 \cdot 3} = c^{27}

\left(d^5\right)^3 = d^{5 \cdot 3} = d^{15}

Combining the Terms

Finally, we can combine the simplified terms to get the final expression:

8 b^{27} c^{27} d^{15}

Conclusion

In this article, we have simplified the given expression using exponents. We applied the power of a product rule to rewrite the expression and then simplified each term using the power of a product rule. The final expression is $8 b^{27} c^{27} d^{15}$ .

Frequently Asked Questions

Q: What is the power of a product rule?

A: The power of a product rule states that when a product of numbers is raised to a power, each number in the product is raised to that power.

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

A: To apply the power of a product rule, simply raise each number in the product to the given power.

Q: What is the final expression after simplifying the given expression?

A: The final expression is $8 b^{27} c^{27} d^{15}$ .

Additional Resources

For more information on exponents and their application, we recommend the following resources:

Final Thoughts

Simplifying expressions using exponents is an essential skill in mathematics. By applying the power of a product rule and simplifying each term, we can rewrite complex expressions in a more compact and simplified form. We hope this article has provided a clear understanding of how to simplify expressions using exponents.

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Frequently Asked Questions

Q: What is the power of a product rule?

A: The power of a product rule states that when a product of numbers is raised to a power, each number in the product is raised to that power. In other words, for any numbers $a$ and $b$ and any positive integer $n$ , we have:

\left(ab\right)^n = a^n b^n

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

A: To apply the power of a product rule, simply raise each number in the product to the given power. For example, if we have the expression $\left(2 b^9 c^9 d^5\right)^3$ , we can rewrite it as:

\left(2 b^9 c^9 d^5\right)^3 = 2^3 \left(b^9\right)^3 \left(c^9\right)^3 \left(d^5\right)^3

Q: What is the difference between a coefficient and an exponent?

A: A coefficient is a number that is multiplied by a variable, while an exponent is a number that is raised to a power. For example, in the expression $3x^2$ , the 3 is a coefficient and the 2 is an exponent.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can use the rule that $a^{-n} = \frac{1}{a^n}$ . For example, if we have the expression $x^{-3}$ , we can rewrite it as:

x^{-3} = \frac{1}{x^3}

Q: What is the zero exponent rule?

A: The zero exponent rule states that any number raised to the power of 0 is equal to 1. In other words, for any number $a$ , we have:

a^0 = 1

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, we can use the rule that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ . For example, if we have the expression $x^{\frac{1}{2}}$ , we can rewrite it as:

x^{\frac{1}{2}} = \sqrt{x}

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two numbers with the same base, we can add their exponents. In other words, for any numbers $a$ and $b$ and any positive integers $m$ and $n$ , we have:

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

Q: How do I simplify expressions with exponents and fractions?

A: To simplify expressions with exponents and fractions, we can use the rules of exponents and fractions. For example, if we have the expression $\frac{x^2}{x^3}$ , we can rewrite it as:

\frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x}

Additional Resources

For more information on exponents and their application, we recommend the following resources:

Final Thoughts

Exponents are a fundamental concept in mathematics, and understanding how to simplify expressions using exponents is essential for success in algebra and beyond. We hope this Q&A article has provided a clear understanding of the rules and concepts of exponents and how to apply them to simplify expressions.