Simplify: { (5y)^3$}$

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression {(5y)^3$}$, and we're asked to simplify it. To do this, we need to apply the rules of exponents, specifically the power of a power rule.

The Power of a Power Rule

The power of a power rule states that when we have an expression in the form ${(a^m)^n}$, we can simplify it by multiplying the exponents. In other words, ${(a^m)^n = a^{m \cdot n}}$. This rule allows us to simplify expressions with multiple exponents.

Applying the Power of a Power Rule

Now, let's apply the power of a power rule to the given expression {(5y)^3$}$. We can see that the expression is in the form ${(a^m)^n}$, where ${a = 5y}$, ${m = 1}$, and ${n = 3}$. Using the power of a power rule, we can simplify the expression as follows:

${(5y)^3 = (5y)^{1 \cdot 3} = 5^{1 \cdot 3} \cdot y^{1 \cdot 3} = 5^3 \cdot y^3}$

Simplifying the Expression

Now that we've applied the power of a power rule, we can simplify the expression further. We can evaluate the exponent ${5^3}$ as follows:

${5^3 = 5 \cdot 5 \cdot 5 = 125}$

So, the simplified expression is:

${5^3 \cdot y^3 = 125y^3}$

Conclusion

In this article, we've simplified the expression {(5y)^3$}$ using the power of a power rule. We've shown that the expression can be simplified as ${125y^3}$. This demonstrates the importance of understanding the rules of exponents and how to apply them to simplify complex expressions.

Additional Examples

Here are a few additional examples of simplifying expressions using the power of a power rule:

• ${(2x)^4 = 2^4 \cdot x^4 = 16x^4}$
• ${(3y)^2 = 3^2 \cdot y^2 = 9y^2}$
• ${(4z)^3 = 4^3 \cdot z^3 = 64z^3}$

These examples demonstrate how the power of a power rule can be used to simplify a wide range of expressions.

Common Mistakes to Avoid

When simplifying expressions using the power of a power rule, there are a few common mistakes to avoid:

• Not applying the rule correctly: Make sure to apply the power of a power rule correctly by multiplying the exponents.
• Not evaluating the exponent: Make sure to evaluate the exponent correctly, as in the case of ${5^3}$.
• Not simplifying the expression further: Make sure to simplify the expression further by evaluating any remaining exponents.

By avoiding these common mistakes, you can ensure that you're simplifying expressions correctly using the power of a power rule.

Final Thoughts

In conclusion, simplifying expressions using the power of a power rule is an essential skill in mathematics. By understanding the rules of exponents and how to apply them, you can simplify complex expressions and solve a wide range of problems. Remember to apply the power of a power rule correctly, evaluate any remaining exponents, and simplify the expression further to ensure that you're getting the correct answer.

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Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about simplifying expressions using the power of a power rule.

Q: What is the power of a power rule?

A: The power of a power rule is a mathematical rule that states that when we have an expression in the form ${(a^m)^n}$, we can simplify it by multiplying the exponents. In other words, ${(a^m)^n = a^{m \cdot n}}$.

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

A: To apply the power of a power rule, simply multiply the exponents. For example, if we have the expression ${(5y)^3}$, we can simplify it by multiplying the exponents as follows:

${(5y)^3 = (5y)^{1 \cdot 3} = 5^{1 \cdot 3} \cdot y^{1 \cdot 3} = 5^3 \cdot y^3}$

Q: What if I have a negative exponent?

A: If you have a negative exponent, you can simplify it by taking the reciprocal of the base and changing the sign of the exponent. For example, if we have the expression ${(5y)^{-3}}$, we can simplify it as follows:

${(5y)^{-3} = (5y)^{(-1) \cdot 3} = (5y)^{-1 \cdot 3} = \frac{1}{5^3 \cdot y^3} = \frac{1}{125y^3}}$

Q: What if I have a fractional exponent?

A: If you have a fractional exponent, you can simplify it by taking the root of the base and changing the sign of the exponent. For example, if we have the expression ${(5y)^{\frac{1}{3}}}$, we can simplify it as follows:

${(5y)^{\frac{1}{3}} = (5y)^{\frac{1}{1 \cdot 3}} = (5y)^{\frac{1}{3}} = \sqrt[3]{5y}}$

Q: Can I simplify expressions with multiple exponents?

A: Yes, you can simplify expressions with multiple exponents by applying the power of a power rule multiple times. For example, if we have the expression ${(5y)^3 \cdot (2x)^2}$, we can simplify it as follows:

${(5y)^3 \cdot (2x)^2 = 5^3 \cdot y^3 \cdot 2^2 \cdot x^2 = 125y^3 \cdot 4x^2 = 500y^3x^2}$

Q: What if I have a variable with a coefficient?

A: If you have a variable with a coefficient, you can simplify it by applying the power of a power rule to the coefficient and the variable separately. For example, if we have the expression ${(3x)^3}$, we can simplify it as follows:

${(3x)^3 = 3^3 \cdot x^3 = 27x^3}$

Q: Can I simplify expressions with negative coefficients?

A: Yes, you can simplify expressions with negative coefficients by applying the power of a power rule to the coefficient and the variable separately. For example, if we have the expression ${(-3x)^3}$, we can simplify it as follows:

${(-3x)^3 = (-3)^3 \cdot x^3 = -27x^3}$

Q: What if I have a variable with a fractional coefficient?

A: If you have a variable with a fractional coefficient, you can simplify it by applying the power of a power rule to the coefficient and the variable separately. For example, if we have the expression ${(\frac{1}{2}x)^3}$, we can simplify it as follows:

${(\frac{1}{2}x)^3 = (\frac{1}{2})^3 \cdot x^3 = \frac{1}{8}x^3}$

Q: Can I simplify expressions with multiple variables?

A: Yes, you can simplify expressions with multiple variables by applying the power of a power rule to each variable separately. For example, if we have the expression ${(5xy)^3}$, we can simplify it as follows:

${(5xy)^3 = 5^3 \cdot x^3 \cdot y^3 = 125x^3y^3}$

Q: What if I have a variable with a negative exponent?

A: If you have a variable with a negative exponent, you can simplify it by taking the reciprocal of the base and changing the sign of the exponent. For example, if we have the expression ${(5xy)^{-3}}$, we can simplify it as follows:

${(5xy)^{-3} = (5xy)^{(-1) \cdot 3} = (5xy)^{-1 \cdot 3} = \frac{1}{5^3 \cdot x^3 \cdot y^3} = \frac{1}{125x^3y^3}}$

Q: Can I simplify expressions with multiple variables and negative exponents?

A: Yes, you can simplify expressions with multiple variables and negative exponents by applying the power of a power rule to each variable and exponent separately. For example, if we have the expression ${(5xy)^{-3}}$, we can simplify it as follows:

${(5xy)^{-3} = (5xy)^{(-1) \cdot 3} = (5xy)^{-1 \cdot 3} = \frac{1}{5^3 \cdot x^3 \cdot y^3} = \frac{1}{125x^3y^3}}$

Q: What if I have a variable with a fractional exponent?

A: If you have a variable with a fractional exponent, you can simplify it by taking the root of the base and changing the sign of the exponent. For example, if we have the expression ${(5xy)^{\frac{1}{3}}}$, we can simplify it as follows:

${(5xy)^{\frac{1}{3}} = (5xy)^{\frac{1}{1 \cdot 3}} = (5xy)^{\frac{1}{3}} = \sqrt[3]{5xy}}$

Q: Can I simplify expressions with multiple variables and fractional exponents?

A: Yes, you can simplify expressions with multiple variables and fractional exponents by applying the power of a power rule to each variable and exponent separately. For example, if we have the expression ${(5xy)^{\frac{1}{3}}}$, we can simplify it as follows:

${(5xy)^{\frac{1}{3}} = (5xy)^{\frac{1}{1 \cdot 3}} = (5xy)^{\frac{1}{3}} = \sqrt[3]{5xy}}$

Q: What if I have a variable with a negative fractional exponent?

A: If you have a variable with a negative fractional exponent, you can simplify it by taking the reciprocal of the base and changing the sign of the exponent. For example, if we have the expression ${(5xy)^{-\frac{1}{3}}}$, we can simplify it as follows:

${(5xy)^{-\frac{1}{3}} = (5xy)^{(-1) \cdot \frac{1}{3}} = (5xy)^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{5xy}}}$

Q: Can I simplify expressions with multiple variables and negative fractional exponents?

A: Yes, you can simplify expressions with multiple variables and negative fractional exponents by applying the power of a power rule to each variable and exponent separately. For example, if we have the expression ${(5xy)^{-\frac{1}{3}}}$, we can simplify it as follows:

${(5xy)^{-\frac{1}{3}} = (5xy)^{(-1) \cdot \frac{1}{3}} = (5xy)^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{5xy}}}$

Conclusion

In this article, we've answered some of the most frequently asked questions about simplifying expressions using the power of a power rule. We've covered a wide range of topics, from simplifying expressions with negative exponents to simplifying expressions with multiple variables and fractional exponents. By understanding the power of a power rule and how to apply it, you can simplify complex expressions and solve a wide range of problems.