Problem 60 Simplify. $$\sqrt{18 y^{6}}+\s... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

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Chapter 8: Problem 60

Simplify. $$\sqrt{18 y^{6}}+\sqrt{32 y^{6}}$$

Short Answer

Expert verified

7 y^3 \sqrt{2}

Step by step solution

Simplify Inside the Square Roots

First, break down the numbers inside the square roots to their prime factors. $ \ \sqrt{18 y^6} = \sqrt{2 \times 3^2 \times y^6} \ \sqrt{32 y^6} = \sqrt{2^5 \times y^6} $

Simplify Using Properties of Radicals

Utilize properties of square roots to separate the components. $ \ \sqrt{18 y^6} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{y^6} = 3 \times y^3 \times \sqrt{2} \ \sqrt{32 y^6} = \sqrt{2^5} \times \sqrt{y^6} = 4 \times y^3 \times \sqrt{2} $

Combine Like Terms

Combine the simplified terms. $ \ 3 y^3 \sqrt{2} + 4 y^3 \sqrt{2} = (3 + 4) y^3 \sqrt{2} = 7 y^3 \sqrt{2} $

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. If you take the square root of 16, you get 4 because 4 * 4 = 16. The symbol for square root is $\text{鈭殅$. In our example, we deal with the square roots of expressions like $\text{鈭�18}y^6$ and $\text{鈭�32}y^6$. To make these simpler, we need to look at the numbers inside the square roots.

Prime Factorization

Prime factorization involves breaking down a number into its prime components, which are prime numbers that multiply together to equal the original number. For example, the prime factors of 18 are 2 and 3 because 2 * 3^2 = 18. Prime numbers are numbers greater than 1 that have no factors other than 1 and themselves. Doing this helps in simplifying square roots. Let's take $\text{鈭�18 y^6}$. We break 18 into prime factors: 2 and 3^2, so we get $\text{鈭�(2 \times 3^2 \times y^6)}$. We do the same for 32 and get $\text{鈭�(2^5 \times y^6)}$.

Properties of Radicals

Radicals, especially square roots, obey certain properties that make them easier to work with. One key property is that the square root of a product $(\text{鈭歛b})$ is the product of the square roots $\text{鈭歛} \times \text{鈭歜}$. For example, $\text{鈭�18 y^6}$ can be broken down to $\text{鈭�2} \times \text{鈭�3^2} \times \text{鈭歽^6}$. Further simplifying, we get $3 y^3 \text{鈭�2}$ because $\text{鈭�3^2} = 3$ and $\text{鈭歽^6} = y^3$. Similarly, $\text{鈭�32 y^6}$ turns into $4 y^3 \text{鈭�2}$, using $\text{鈭�2^5} = 4 \text{鈭�2}$ and $\text{鈭歽^6} = y^3$.

Combining Like Terms

In algebra, you often combine like terms to simplify an expression. Like terms are terms whose variables and exponents are the same. For the terms $3 y^3 \text{鈭�2}$ and $4 y^3 \text{鈭�2}$, the variable part $y^3 \text{鈭�2}$ is the same in both. So you add the coefficients 3 and 4 to get 7. The final step in simplifying $\text{鈭�18 y^6} + \text{鈭�32 y^6}$ is combining these like terms to get $7 y^3 \text{鈭�2}$. This makes our simplified form of the original expression.

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91影视

Simplify. $$\sqrt{18 y^{6}}+\sqrt{32 y^{6}}$$

Short Answer

Step by step solution

Simplify Inside the Square Roots

Simplify Using Properties of Radicals

Combine Like Terms

Key Concepts

Square Roots

Prime Factorization

Properties of Radicals

Combining Like Terms

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