Simplify: $\left(3y^2\right)^3$

$$

Understanding the Problem

Exponentiation is a mathematical operation that involves raising a number or an expression to a power. In this problem, we are given the expression $\left(3y^2\right)^3$ and we need to simplify it. To simplify an expression with exponents, we need to apply the rules of exponentiation.

Rules of Exponentiation

When we have an expression with exponents, we can simplify it by applying the following rules:

• Product of Powers Rule: When we multiply two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When we raise a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Simplifying the Expression

Now, let's apply the rules of exponentiation to simplify the expression $\left(3y^2\right)^3$.

Using the Power of a Power Rule, we can rewrite the expression as:

$$\left(3y^2\right)^3 = 3^3 \cdot (y^2)^3$$

Now, we can apply the Product of Powers Rule to simplify the expression further:

$$3^3 \cdot (y^2)^3 = 3^3 \cdot y^{2 \cdot 3}$$

Using the Power of a Power Rule again, we can rewrite the expression as:

$$3^3 \cdot y^{2 \cdot 3} = 3^3 \cdot y^6$$

Now, we can simplify the expression further by evaluating the exponent:

$$3^3 \cdot y^6 = 27y^6$$

Therefore, the simplified expression is $\boxed{27y^6}$.

Conclusion

In this article, we learned how to simplify an expression with exponents using the rules of exponentiation. We applied the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule to simplify the expression $\left(3y^2\right)^3$. We also evaluated the exponent to get the final simplified expression, which is $\boxed{27y^6}$.

Frequently Asked Questions

• What is exponentiation? Exponentiation is a mathematical operation that involves raising a number or an expression to a power.
• What are the rules of exponentiation? The rules of exponentiation are the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule.
• How do I simplify an expression with exponents? To simplify an expression with exponents, you need to apply the rules of exponentiation, such as the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule.

Example Problems

• Simplify the expression $\left(2x^3\right)^2$. Using the Power of a Power Rule, we can rewrite the expression as $2^2 \cdot (x^3)^2$. Then, using the Product of Powers Rule, we can simplify the expression further: $2^2 \cdot (x^3)^2 = 2^2 \cdot x^{3 \cdot 2} = 4x^6$.
• Simplify the expression $\left(5y^4\right)^3$. Using the Power of a Power Rule, we can rewrite the expression as $5^3 \cdot (y^4)^3$. Then, using the Product of Powers Rule, we can simplify the expression further: $5^3 \cdot (y^4)^3 = 5^3 \cdot y^{4 \cdot 3} = 125y^{12}$.

Further Reading

• Exponentiation Rules: This article provides a comprehensive overview of the rules of exponentiation, including the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule.
• Simplifying Expressions with Exponents: This article provides step-by-step instructions on how to simplify expressions with exponents using the rules of exponentiation.
• Exponentiation Examples: This article provides a collection of example problems that demonstrate how to simplify expressions with exponents using the rules of exponentiation.
  

Understanding Exponentiation

Exponentiation is a mathematical operation that involves raising a number or an expression to a power. It is a fundamental concept in mathematics that is used to describe the repeated multiplication of a number or expression. In this article, we will answer some of the most frequently asked questions about exponentiation.

Q&A

Q: What is exponentiation?

A: Exponentiation is a mathematical operation that involves raising a number or an expression to a power. It is a fundamental concept in mathematics that is used to describe the repeated multiplication of a number or expression.

Q: What are the rules of exponentiation?

A: The rules of exponentiation are the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule. These rules are used to simplify expressions with exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponentiation, such as the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule.

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

A: A power is the result of raising a number or expression to a power, while an exponent is the number or expression that is being raised to a power.

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

A: Yes, you can simplify an expression with a negative exponent by using the Negative Exponent Rule, which states that $a^{-n} = \frac{1}{a^n}$.

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

A: Yes, you can simplify an expression with a fractional exponent by using the Fractional Exponent Rule, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you need to apply the rules of exponentiation and simplify the expression.

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents by using the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule.

Q: How do I simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you need to apply the Zero Exponent Rule, which states that $a^0 = 1$.

Example Problems

• Simplify the expression $\left(2x^3\right)^2$. Using the Power of a Power Rule, we can rewrite the expression as $2^2 \cdot (x^3)^2$. Then, using the Product of Powers Rule, we can simplify the expression further: $2^2 \cdot (x^3)^2 = 2^2 \cdot x^{3 \cdot 2} = 4x^6$.
• Simplify the expression $\left(5y^4\right)^3$. Using the Power of a Power Rule, we can rewrite the expression as $5^3 \cdot (y^4)^3$. Then, using the Product of Powers Rule, we can simplify the expression further: $5^3 \cdot (y^4)^3 = 5^3 \cdot y^{4 \cdot 3} = 125y^{12}$.

Further Reading

• Exponentiation Rules: This article provides a comprehensive overview of the rules of exponentiation, including the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule.
• Simplifying Expressions with Exponents: This article provides step-by-step instructions on how to simplify expressions with exponents using the rules of exponentiation.
• Exponentiation Examples: This article provides a collection of example problems that demonstrate how to simplify expressions with exponents using the rules of exponentiation.

Common Mistakes

• Forgetting to apply the rules of exponentiation: When simplifying expressions with exponents, it is easy to forget to apply the rules of exponentiation. Make sure to apply the Product of Powers Rule, Power of a Power Rule, and Zero Exponent Rule to simplify the expression.
• Not evaluating the exponent: When simplifying expressions with exponents, it is easy to forget to evaluate the exponent. Make sure to evaluate the exponent to get the final simplified expression.
• Not checking the final answer: When simplifying expressions with exponents, it is easy to make mistakes. Make sure to check the final answer to ensure that it is correct.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponentiation. We have covered the rules of exponentiation, how to simplify expressions with exponents, and how to evaluate expressions with exponents. We have also provided example problems and further reading to help you learn more about exponentiation.