Breaking Down the Cube Roots

Oct 13, 2025 by leolavanderia.com.br 147 views

Understanding the Problem

The given problem involves finding the sum of two cube roots, each containing variables x and y raised to certain powers. To solve this problem, we need to apply the properties of radicals and exponents. The expression to be evaluated is:

$\sqrt[3]{125 x^{10} y^{13}} + \sqrt[3]{27 x^{10} y^{13}}$

Breaking Down the Cube Roots

We can start by breaking down the cube roots into their respective components. The cube root of a number can be expressed as the number raised to the power of 1/3. Therefore, we can rewrite the given expression as:

$\sqrt[3]{125 x^{10} y^{13}} = (125 x^{10} y^{13})^{\frac{1}{3}}$

$\sqrt[3]{27 x^{10} y^{13}} = (27 x^{10} y^{13})^{\frac{1}{3}}$

Simplifying the Cube Roots

Now, we can simplify the cube roots by applying the property of exponents that states $(a^m)^n = a^{mn}$ . Using this property, we can rewrite the cube roots as:

$(125 x^{10} y^{13})^{\frac{1}{3}} = 125^{\frac{1}{3}} x^{\frac{10}{3}} y^{\frac{13}{3}}$

$(27 x^{10} y^{13})^{\frac{1}{3}} = 27^{\frac{1}{3}} x^{\frac{10}{3}} y^{\frac{13}{3}}$

Evaluating the Cube Roots

Next, we need to evaluate the cube roots of the numbers 125 and 27. The cube root of 125 is 5, and the cube root of 27 is 3. Therefore, we can rewrite the expression as:

$5 x^{\frac{10}{3}} y^{\frac{13}{3}} + 3 x^{\frac{10}{3}} y^{\frac{13}{3}}$

Combining Like Terms

Now, we can combine the like terms in the expression. Since both terms have the same variables raised to the same powers, we can add their coefficients. Therefore, we can rewrite the expression as:

$(5 + 3) x^{\frac{10}{3}} y^{\frac{13}{3}}$

Simplifying the Expression

Finally, we can simplify the expression by evaluating the sum of the coefficients. The sum of 5 and 3 is 8. Therefore, we can rewrite the expression as:

$8 x^{\frac{10}{3}} y^{\frac{13}{3}}$

Rationalizing the Expression

To rationalize the expression, we need to get rid of the fractional exponents. We can do this by multiplying the expression by a suitable form of 1. In this case, we can multiply the expression by $x^{-\frac{10}{3}} y^{-\frac{13}{3}}$ . This will eliminate the fractional exponents and give us a rationalized expression.

$8 x^{\frac{10}{3}} y^{\frac{13}{3}} \cdot x^{-\frac{10}{3}} y^{-\frac{13}{3}}$

Simplifying the Rationalized Expression

Now, we can simplify the rationalized expression by applying the property of exponents that states $a^m \cdot a^n = a^{m+n}$ . Using this property, we can rewrite the expression as:

$8 x^{\frac{10}{3} - \frac{10}{3}} y^{\frac{13}{3} - \frac{13}{3}}$

Evaluating the Exponents

Next, we need to evaluate the exponents in the expression. The exponent of x is $\frac{10}{3} - \frac{10}{3}$ , which equals 0. The exponent of y is $\frac{13}{3} - \frac{13}{3}$ , which also equals 0. Therefore, we can rewrite the expression as:

$8 x^0 y^0$

Simplifying the Expression

Finally, we can simplify the expression by applying the property of exponents that states $a^0 = 1$ . Using this property, we can rewrite the expression as:

$8 \cdot 1 \cdot 1$

Evaluating the Expression

Now, we can evaluate the expression by multiplying the numbers. The product of 8 and 1 is 8. Therefore, we can rewrite the expression as:

$8$

Rationalizing the Expression

To rationalize the expression, we need to get rid of the fractional exponents. We can do this by multiplying the expression by a suitable form of 1. In this case, we can multiply the expression by $(\sqrt[3]{x y})^3$ . This will eliminate the fractional exponents and give us a rationalized expression.

$8 \cdot (\sqrt[3]{x y})^3$

Simplifying the Rationalized Expression

Now, we can simplify the rationalized expression by applying the property of exponents that states $(a^m)^n = a^{mn}$ . Using this property, we can rewrite the expression as:

$8 x^3 y^4$

Evaluating the Expression

Now, we can evaluate the expression by multiplying the numbers. The product of 8 and 1 is 8. Therefore, we can rewrite the expression as:

$8 x^3 y^4(\sqrt[3]{x y})$

The final answer is: $\boxed{8 x^3 y^4(\sqrt[3]{x y})}$

Understanding the Problem

$\sqrt[3]{125 x^{10} y^{13}} + \sqrt[3]{27 x^{10} y^{13}}$

Q&A

Q: What is the first step in solving this problem?

A: The first step in solving this problem is to break down the cube roots into their respective components. We can rewrite the given expression as:

$\sqrt[3]{125 x^{10} y^{13}} = (125 x^{10} y^{13})^{\frac{1}{3}}$

$\sqrt[3]{27 x^{10} y^{13}} = (27 x^{10} y^{13})^{\frac{1}{3}}$

Q: How do we simplify the cube roots?

A: We can simplify the cube roots by applying the property of exponents that states $(a^m)^n = a^{mn}$ . Using this property, we can rewrite the cube roots as:

$(125 x^{10} y^{13})^{\frac{1}{3}} = 125^{\frac{1}{3}} x^{\frac{10}{3}} y^{\frac{13}{3}}$

$(27 x^{10} y^{13})^{\frac{1}{3}} = 27^{\frac{1}{3}} x^{\frac{10}{3}} y^{\frac{13}{3}}$

Q: What are the cube roots of 125 and 27?

A: The cube root of 125 is 5, and the cube root of 27 is 3. Therefore, we can rewrite the expression as:

$5 x^{\frac{10}{3}} y^{\frac{13}{3}} + 3 x^{\frac{10}{3}} y^{\frac{13}{3}}$

Q: How do we combine like terms?

A: We can combine the like terms in the expression by adding their coefficients. Therefore, we can rewrite the expression as:

$(5 + 3) x^{\frac{10}{3}} y^{\frac{13}{3}}$

Q: What is the final simplified expression?

A: The final simplified expression is:

$8 x^{\frac{10}{3}} y^{\frac{13}{3}}$

Q: How do we rationalize the expression?

A: To rationalize the expression, we need to get rid of the fractional exponents. We can do this by multiplying the expression by a suitable form of 1. In this case, we can multiply the expression by $x^{-\frac{10}{3}} y^{-\frac{13}{3}}$ . This will eliminate the fractional exponents and give us a rationalized expression.

$8 x^{\frac{10}{3}} y^{\frac{13}{3}} \cdot x^{-\frac{10}{3}} y^{-\frac{13}{3}}$

Q: What is the final rationalized expression?

A: The final rationalized expression is:

$8 x^3 y^4$

Q: How do we evaluate the expression?

A: We can evaluate the expression by multiplying the numbers. The product of 8 and 1 is 8. Therefore, we can rewrite the expression as:

$8 x^3 y^4(\sqrt[3]{x y})$

Conclusion

The final answer to the problem is $\boxed{8 x^3 y^4(\sqrt[3]{x y})}$ . This solution involves breaking down the cube roots, simplifying the expression, rationalizing the expression, and evaluating the final expression.